Celebratio Mathematica

The start­ing point of the pa­per [1] was Thur­ston’s amaz­ing in­sight in the 1980’s that all 3-di­men­sion­al man­i­folds should be ca­non­ic­ally di­vis­ible in­to pieces hav­ing a well-defined geo­met­ric struc­ture of one of eight types, the most im­port­ant of which is the hy­per­bol­ic one. In con­junc­tion with the fam­ous Mostow ri­gid­ity the­or­em, this means that 3-di­men­sion­al to­po­logy be­comes a part, first of dif­fer­en­tial geo­metry, and then of al­geb­ra­ic num­ber the­ory, something that is not at all the case in oth­er di­men­sions. The main class is that of ori­ented hy­per­bol­ic 3-man­i­folds, which have a rieman­ni­an met­ric with con­stant neg­at­ive curvature that can be nor­mal­ized to $-1$ and hence are loc­ally iso­met­ric to hy­per­bol­ic 3-space ${\mathbb{H}}^3$. Of par­tic­u­lar in­terest is the volume spec­trum, the set of volumes of all com­plete hy­per­bol­ic 3-man­i­folds of fi­nite volume (in which case they are either com­pact or the uni­on of a com­pact part and a fi­nite num­ber of “cusps” dif­feo­morph­ic to the product of a half-line and a tor­us). These volumes have both strik­ing num­ber-the­or­et­ic­al prop­er­ties (they be­long to the im­age un­der the reg­u­lat­or map of the Bloch group, as dis­cussed in Sec­tion 3) and strik­ing met­ric prop­er­ties (they form a count­able well-ordered sub­set of ${\mathbb{R}}_{>0}$, as dis­cussed in Sec­tion 2), and the primary goal of the pa­per [1] was to un­der­stand them as thor­oughly as pos­sible.

In this sec­tion we dis­cuss ideal tri­an­gu­la­tions and their NZ-equa­tions in some de­tail. Ideal tri­an­gu­la­tions of 3-man­i­folds with tor­us bound­ary com­pon­ents were in­tro­duced by Thur­ston [e7] as a con­veni­ent way to de­scribe and ef­fect­ively com­pute [e24] com­plete hy­per­bol­ic struc­tures on 3-man­i­folds. Re­call that in hy­per­bol­ic geo­metry an ideal tet­ra­hed­ron is the con­vex hull of four points in the bound­ary $\partial ({\mathbb{H}}^3)\cong {\mathbb{P}}^1(\mathbb{C})$ of 3-di­men­sion­al hy­per­bol­ic space ${\mathbb{H}}^3$. The (ori­ent­a­tion-pre­serving) iso­metry group of ${\mathbb{H}}^3$ is the group ${\mathrm{PSL}}_2(\mathbb{C})$, act­ing on the bound­ary by frac­tion­al lin­ear trans­form­a­tions of ${\mathbb{P}}^1(\mathbb{C})$, and since un­der this group any four dis­tinct points can be put in stand­ard po­s­i­tion $(0,1,\mathrm{\infty },z)$ for some $z\in \mathbb{C}\setminus \{0,1\}$ (the cross-ra­tio of the four points), any (ori­ented) ideal tet­ra­hed­ron is the con­vex hull $\mathrm{\Delta }(z)$ for some com­plex num­ber $z\in \mathbb{C}\setminus \{0,1\}$, called the shape para­met­er of the tet­ra­hed­ron. This num­ber is not quite unique be­cause of the choice of which three ver­tices we send to 0, 1 and $\mathrm{\infty }$, mean­ing that the ori­ented tet­ra­hedra $\mathrm{\Delta }(z)$, $\mathrm{\Delta }(z^{\prime })$ and $\mathrm{\Delta }(z^{\prime \prime })$, where $z^{\prime }=1/(1-z)$ and $z^{\prime \prime }=1-1/z$, are iso­met­ric. In this way one at­taches a shape para­met­er $z$, $z^{\prime }$ or $z^{\prime \prime }$ to each pair of op­pos­ite edges of a giv­en ori­ented ideal tet­ra­hed­ron. Note that the shape of a tet­ra­hed­ron is an ar­bit­rary com­plex num­ber not equal to 0 or 1, and wheth­er it has pos­it­ive, zero or neg­at­ive ima­gin­ary part is ir­rel­ev­ant to our dis­cus­sion.

When $\mathcal{T}$ is an ideal tri­an­gu­la­tion of a hy­per­bol­ic 3-man­i­fold $M$ with $N$ tet­ra­hedra with shape para­met­ers $z_1,\dots ,z_N$, then we get one poly­no­mi­al equa­tion (“glu­ing equa­tion”) for each edge. Spe­cific­ally, the shape para­met­ers at that edge of all tet­ra­hedra that are in­cid­ent with it must clearly have ar­gu­ments that add up to $2\pi$ (be­cause oth­er­wise the met­ric would not be smooth along that edge), but in fact the shape para­met­ers them­selves have product $+1$ by an easy ar­gu­ment. Since each of the pos­sible shape para­met­ers $z$, $z^{\prime }$, $z^{\prime \prime }$ of $\mathrm{\Delta }(z)$ be­longs to the mul­ti­plic­at­ive group $⟨z,1-z,-1⟩$, this equa­tion for each edge $e_i$ has the form $$\begin{array}{}\text{(1)}& \pm \prod _{j=1}^N{z}_j^{R_{ij}^{\prime }}(1-z_j{)}^{R_{ij}^{\prime \prime }}=1.\end{array}$$

Since it is eas­ily seen that the num­ber of edges in the tri­an­gu­la­tion is the same as the num­ber $N$ of sim­plices, this gives us $N$ poly­no­mi­al equa­tions among the $N$ com­plex num­bers $z_1,\dots ,z_N$. The ob­vi­ous thought is that this leads to a 0-di­men­sion­al mod­uli space and ex­plains the ri­gid­ity, but this is wrong since ri­gid­ity only ap­plies to com­plete hy­per­bol­ic struc­tures, and in fact the $N$ edge re­la­tions are nev­er mul­ti­plic­at­ively in­de­pend­ent. An ob­vi­ous ex­ample is that their product is al­ways 1, and this is the only de­pend­ence if the bound­ary of the 3-man­i­fold is a tor­us (of­ten called a cusp), but in gen­er­al there are $h$ mul­ti­plic­at­ively in­de­pend­ent re­la­tions among the $N$ equa­tions (1), where $h$ is the num­ber of cusps of the 3-man­i­fold (as­sum­ing that all bound­ary com­pon­ents are tori). Thus the true ex­pec­ted di­men­sion of the mod­uli space of hy­per­bol­ic struc­tures is in fact $h$. That the di­men­sion really is $h$ is an im­port­ant the­or­em of Thur­ston (§5 of [e7]) and was giv­en a new and sim­pler proof in [1] us­ing the al­geb­ra­ic struc­ture of the glu­ing equa­tions. The 0-di­men­sion­al mod­uli space (ri­gid­ity) arises when we re­quire the hy­per­bol­ic struc­ture giv­en by the shape para­met­ers $z_i$ to be com­plete, be­cause this leads to two fur­ther in­de­pend­ent re­la­tions at each cusp. Spe­cific­ally, each “peri­pher­al curve” (mean­ing an iso­topy class of curves on the tor­us cross-sec­tion of one of the cusps) gives a re­la­tion, so if one chooses a me­ridi­an and a lon­git­ude $({\mu }_i,{\lambda }_i)$ at each cusp $i=1,\dots ,h$, one ob­tains $2h$ fur­ther equa­tions $$\begin{array}{}\text{(2)}& \pm \prod _{j=1}^N{z}_j^{M_{ij}^{\prime }}(1-z_j{)}^{M_{ij}^{\prime \prime }}=1,\pm \prod _{j=1}^N{z}_j^{L_{ij}^{\prime }}(1-z_j{)}^{L_{ij}^{\prime \prime }}=1(i=1,\dots ,h).\end{array}$$ Thus the full set of glu­ing equa­tions is de­scribed by an $(N+2h)\times 2N$ mat­rix $$U=\left(\begin{array}{cc}{R}^{\prime }& R^{\prime \prime }\\ M^{\prime }& M^{\prime \prime }\\ L^{\prime }& L^{\prime \prime }\end{array}\right).$$

It was shown in [1] that the mat­rix $U$ has some key sym­plect­ic prop­er­ties, of which the most im­port­ant (The­or­em 2.2) says that $$\begin{array}{}\text{(3)}& U{J}_{2N}{U}^t=(\begin{array}{cc}{0}_{N,N}& 0_{2h,N}\\ 0_{N,2h}& 2J_{2h}\end{array}),\end{array}$$ where $$J_{2n}:=(\begin{array}{cc}0\phantom{{}_n}& -1_n\\ 1_n& \phantom{-}0\phantom{{}_n}\end{array}),$$ while the oth­ers say that the $N\times 2N$ mat­rix $R=(R^{\prime }{R}^{\prime \prime })$ has rank $N-h$, the full mat­rix $U$ has rank $N+h$, and the space $[U]\subset {\mathbb{R}}^{2N}$ spanned by the rows of $U$ is the or­tho­gon­al com­ple­ment of $[R]$ with re­spect to the sym­plect­ic struc­ture $J_{2N}$ (Pro­pos­i­tion 2.3). The rank state­ment was then used to prove that the di­men­sion of the above-men­tioned de­form­a­tion space of (non­com­plete) hy­per­bol­ic struc­tures is at least $h$, and then a fur­ther ar­gu­ment us­ing Mostow ri­gid­ity showed that it is ex­actly $h$. This gives rise in the 1-cusp case to a poly­no­mi­al $P(m,\ell )=0$, where $m^2$ and ${\ell }^2$ de­note the left-hand sides of the two equa­tions (2) and $P$ is a cer­tain poly­no­mi­al (a factor of what is now called the $A$-poly­no­mi­al) that was cal­cu­lated ex­pli­citly in [1] for the simplest hy­per­bol­ic knot $4_1$ (fig­ure-8). In the gen­er­al case the de­form­a­tion space is a com­pon­ent of the “char­ac­ter vari­ety” as in­tro­duced and stud­ied in [e4] and stud­ied later by Wal­ter and his stu­dents Ab­hijit Cham­pan­erkar [e11] and Stefan Till­mann [e12].

To get a com­plete man­i­fold with $h$ cusps, we im­pose $h$ fur­ther glu­ing con­di­tions, namely the equa­tions (1) and the first of each of the equa­tions (2). (But in fact these $h$ me­ridi­an equa­tions to­geth­er with the edge equa­tions im­ply the lon­git­ude equa­tions, so in the end all re­la­tions (1) and (2) hold.) Now ri­gid­ity ap­plies and there are no de­form­a­tions. But there is an­oth­er pro­cess, with a lot more free­dom, to ob­tain ri­gid com­plete hy­per­bol­ic struc­tures from the ori­gin­al cusped man­i­fold, namely to do a Dehn sur­gery at some or all of the cusps. Spe­cific­ally, when we do a $(p,q)$-sur­gery at a cusp (mean­ing that we trun­cate the 3-man­i­fold at the tor­us bound­ary and glue on a sol­id tor­us in such a way as to kill the ho­mo­topy class of $p$ times a chosen me­ridi­an times $q$ times a chosen lon­git­ude), then we im­pose the glu­ing equa­tion $m^{2p}{\ell }^{2q}=1$, where $m^2$ and ${\ell }^2$ as above are the left-hand sides of one of the pairs of equa­tions (2). In the next sec­tion, still fol­low­ing [1], we dis­cuss how the volumes be­have un­der this pro­cess.

As already stated, the ori­gin­al main pur­pose of [1] was to study the arith­met­ic of the set of volumes of all hy­per­bol­ic 3-man­i­folds. It had been ob­served by Thur­ston, us­ing earli­er work of Jørgensen, that this volume spec­trum is a well-ordered sub­set of the pos­it­ive reals. In oth­er words, there is a smal­lest volume (which is known), a second smal­lest, a third smal­lest, …, then a smal­lest lim­it point, a second smal­lest lim­it point, …, then lim­its of these, and so on. The proof of well-ordered­ness shows that these simple and high­er-or­der lim­it points arise by “clos­ing up” one or more of the cusps of a non­com­pact hy­per­bol­ic 3-man­i­fold $M$ by Dehn sur­ger­ies to ob­tain a count­able col­lec­tion of man­i­folds with few­er cusps whose volumes tend from be­low to that of $M$. The ob­ject of [1] was to study the speed with which these volumes con­verge.

Be­fore study­ing the ef­fect of sur­ger­ies, we must un­der­stand the volume of a single hy­per­bol­ic 3-man­i­fold $M$. Clearly it can be giv­en as the sum of the volumes of the tet­ra­hedra of any ideal tri­an­gu­la­tion, so the first step is to un­der­stand these. An (ori­ented and nonde­gen­er­ate) ideal tet­ra­hed­ron can be para­met­rized either by a shape para­met­er $z$ in the com­plex up­per half-plane, as ex­plained in the pre­ced­ing sec­tion, or, in case $z$ is in the up­per half-plane, by the three angles $\alpha$, $\beta$, $\gamma$ (pos­it­ive and with $\alpha +\beta +\gamma =\pi$) of the Eu­c­lidean tri­angle that one “sees” by look­ing at the tet­ra­hed­ron from any of its four cusps. When $\mathrm{Im}(z)>0$, the angles and the shapes are re­lated by $(\alpha ,\beta ,\gamma )=(\arg (z),\arg (z^{\prime }),\arg (z^{\prime \prime }))$, with $z^{\prime }$ and $z^{\prime \prime }$ as above.

Ac­cord­ing to Chapter 7 of Thur­ston’s notes, writ­ten by Mil­nor, the volume of this tet­ra­hed­ron when $\mathrm{Im}(z)>0$ is giv­en in terms of these two para­met­riz­a­tions by the two for­mu­las $$\begin{array}{}\text{(4)}& \mathrm{Vol}(\mathrm{\Delta }(z))=D(z)=\text{Л}(\alpha )+\text{Л}(\beta )+\text{Л}(\gamma ),\end{array}$$ the equal­ity of the two be­ing an iden­tity due to Kum­mer. (Both for­mu­las are ac­tu­ally true for all $z\ne 0,1$, but the sum of $\alpha$, $\beta$ and $\gamma$ is $-\pi$ if $z$ is in the lower half-plane.) Here $D(z)$ and $\text{Л}(\theta )$ are the Bloch–Wign­er di­log­ar­ithm (see Sec­tion 3) and the Lob­achevsky func­tion, defined re­spect­ively by $$\begin{array}{}\text{(5)}& \begin{array}{rl}D(z)& =\mathrm{Im}({\mathrm{Li}}_2(z)+\log |z|\log (1-z)),\\ \text{Л}(\theta )& =-{\int }_0^{\theta }\log |2\sin t|dt=\frac{1}{2}\sum _{n=1}^{\mathrm{\infty }}\frac{\sin (2n\theta )}{n^2}=\frac{1}{2}D(e^{2i\theta }).\end{array}\end{array}$$ Thus the volume of a hy­per­bol­ic man­i­fold $M$ tri­an­gu­lated by $N$ ideal tet­ra­hedra with shape para­met­ers $z_i$ is giv­en in terms of the di­log­ar­ithm func­tion by $$\begin{array}{}\text{(6)}& \mathrm{Vol}(M)=\sum _{j=1}^{N}D(z_j).\end{array}$$

Now let $M$ be a 3-man­i­fold with $h$ cusps. It has a unique com­plete hy­per­bol­ic struc­ture as a quo­tient ${\mathbb{H}}^3/\mathrm{\Gamma }$ for some lat­tice $\mathrm{\Gamma }\subset {\mathrm{PSL}}_2(\mathbb{C})$ iso­morph­ic to ${\pi }_1(M)$, the cor­res­pond­ing shape para­met­ers $z^0=(z_1^0,\dots ,z_N^0)$ be­ing a solu­tion of all $N+2h$ equa­tions (1) and (2). A small de­form­a­tion of this hy­per­bol­ic struc­ture will have a nearby shape para­met­er vec­tor $z=(z_1,\dots ,z_N)$ sat­is­fy­ing only (1). By the res­ult of Thur­ston men­tioned earli­er, the space of all such de­form­a­tions has the struc­ture of a smooth com­plex man­i­fold of di­men­sion $h$, so it is iso­morph­ic to a small neigh­bor­hood $\mathfrak{U}$ of 0 in ${\mathbb{C}}^h$. Thus each $z_i$ de­pends holo­morph­ic­ally on $\mathfrak{u}\in \mathfrak{U}$ and $z_i(0)=z_i^0$. To make this more vis­ible, and to choose nice co­ordin­ates ${\mathfrak{u}}_i$ on $\mathfrak{U}$, we have to look more closely at the struc­ture of the cusps on $M$ and at Dehn sur­ger­ies.

Each cusp has the struc­ture $[0,\mathrm{\infty })\times T^2$, where $T^2$ is a totally geodesic­ally em­bed­ded tor­us in $M$ and has a flat Eu­c­lidean met­ric, unique up to ho­mothety, in­duced by the hy­per­bol­ic met­ric on $M$. More con­cretely, the cusps of $M={\mathbb{H}}^3/\mathrm{\Gamma }$ are in­dexed by $P\in {\mathbb{P}}^1(\mathbb{C})$, whose sta­bil­izer ${\mathrm{\Gamma }}_P$ is free abeli­an of rank 2. After con­jug­a­tion we can place $P$ at $\mathrm{\infty }$, in which case ${\mathrm{\Gamma }}_{\mathrm{\infty }}$ has the form $$\pm (\begin{array}{cc}1& \mathrm{\Lambda }\\ 0& 1\end{array})$$ for some lat­tice $\mathrm{\Lambda }\subset \mathbb{C}$, unique up to ho­mothety, and then we can identi­fy $\mathrm{\Lambda }$ with $H_1(T^2)$ and $T^2$ with $\mathbb{C}/\mathrm{\Lambda }$. There is a ca­non­ic­al quad­rat­ic form on $\mathrm{\Lambda }$ defined as the square of the length of a vec­tor di­vided by the volume of $T^2$. If we choose an ori­ented basis $(\mu ,\lambda )$ of $H_1(T^2)$ (“me­ridi­an” and “lon­git­ude”) to identi­fy $\mathrm{\Lambda }$ with ${\mathbb{Z}}^2$, then this quad­rat­ic form is giv­en by $$Q(p,q)=\frac{|p\tau +q{|}^2}{\mathrm{Im}(\tau )}$$ if we res­cale $\mathrm{\Lambda }$ after ho­mothety to be $\mathbb{Z}\tau +\mathbb{Z}$ for some $\tau$ in the com­plex up­per half-plane.

On the oth­er hand, each ele­ment of $\mathrm{\Lambda }=H_1(T^2)$ can be iden­ti­fied with an iso­topy class of closed curves on $T^2$. Do­ing a $(p,q)$ Dehn sur­gery at a cusp, once the basis of $H_1(T^2)$ has been chosen, means re­mov­ing $T^2\times [0,\mathrm{\infty })$ from $M$ and re­pla­cing it by a sol­id tor­us in such way that the curve on $T^2$ cor­res­pond­ing to the class $p\mu +q\lambda$ bounds in the sol­id tor­us. Here we as­sume that the in­tegers $p$ and $q$ are coprime, but we also al­low “$\mathrm{\infty }$” as a value for $(p,q)$, mean­ing that we leave this cusp un­touched. The Dehn sur­ger­ies on $M$ are then de­scribed by tuples $$\kappa =((p_1,q_1),\dots ,(p_h,q_h))\in ({\mathbb{Z}}^2\cup \{\mathrm{\infty }\}{)}^h.$$ Thur­ston’s the­or­em [e7] tells us that if all $(p_i,q_i)$ are near enough to $\mathrm{\infty }$ (mean­ing that $(p_i,q_i)=\mathrm{\infty }$ or $p_i^2+q_i^2$ is large), the surgered man­i­fold $M_{\kappa }$ is hy­per­bol­ic, in which case both the shape para­met­ers $z_1,\dots ,z_N$ and the volume $\mathrm{Vol}(M_{\kappa })$ be­come func­tions of $\kappa$. Ex­pli­citly, the de­formed shape para­met­ers $\mathit{z}(\kappa )=\mathit{z}(\mathit{p},\mathit{b}\mathit{q})$ that tend to ${\mathit{z}}^0$ as all the pairs $(p_i,q_i)$ tend to in­fin­ity are giv­en by adding to the ori­gin­al glu­ing equa­tions (1) the $h$ new equa­tions giv­en by the product of the $p_i$-th power of the first ex­pres­sion by the $q_i$-th power of the second one in (2), and then $\mathrm{Vol}(M_{\kappa })$ is giv­en by (6) with $z_j$ re­placed by $z_j(\kappa )$.

We now get two real num­bers at each cusp: the value $Q_i(p_i,q_i)$ of the quad­rat­ic form cor­res­pond­ing to that cusp at the pair $(p_i,q_i)$ (with the con­ven­tion $Q_i(\mathrm{\infty })=\mathrm{\infty }$) and the length $L_i=L_i(\kappa )$ of the short geodes­ic on $M_{\kappa }$, which is the core of the sol­id tor­us ad­ded by the Dehn sur­gery (or 0 if the $i$-th cusp has not been surgered). They are re­lated by $$\begin{array}{}\text{(7)}& L_i=\frac{2\pi }{Q_i(p_i,q_i)}+\mathrm{O}(\sum _{i=1}^h\frac{1}{p_i^4+q_i^4})\end{array}$$ ([1], Pro­pos­i­tion 4.3) as all ${\kappa }_i=(p_i,q_i)$ tend to in­fin­ity. The main volume res­ult of [1], proved by us­ing (6) and ana­lyz­ing the changes of the di­log­ar­ithms un­der small changes of the $z$’s, is then giv­en by the pair of asymp­tot­ic for­mu­las $$\begin{array}{}\text{(8a)}& \mathrm{Vol}(M_{\kappa })& =\mathrm{Vol}(M)-\sum _{i=1}^h(\frac{{\pi }^2}{Q_i(p_i,q_i)}+\mathrm{O}(\frac{1}{p_i^4+q_i^4}))\text{(8b)}& & =\mathrm{Vol}(M)-\sum _{i=1}^h(\frac{\pi L_i}{2}+\mathrm{O}(L_i^2))\end{array}$$ (The­or­ems 1A and 1B in [1]), which are equi­val­ent to one an­oth­er by vir­tue of (7). These volumes, as $\kappa =(\mathit{p},\mathit{b}\mathit{q})$ ranges over all $h$-tuples of suf­fi­ciently large pairs of coprime in­tegers or the sym­bol $\mathrm{\infty }$ mean­ing un­surgered, all be­long to the hy­per­bol­ic volume spec­trum, and equa­tion (8a) has as an im­me­di­ate co­rol­lary a de­scrip­tion of the loc­al struc­ture of this volume spec­trum near its lim­it point, be­cause the asymp­tot­ics of the num­ber of lat­tice points, or of prim­it­ive lat­tice points, in a large el­lipse, is well known. The pre­cise asymp­tot­ic state­ment, which we will not re­peat here, is for­mu­lated ex­pli­citly as a co­rol­lary to The­or­em 1A in [1].

All of this is only for in­teg­ral (and coprime) val­ues of $p_i$ and $q_i$. However, as is ex­plained in [1] in de­tail, the shape para­met­ers $z_j(\kappa )$ and the lengths $L_i(\kappa )$ are defined for $p$ and $q$ real rather than just in­teg­ral and coprime in pairs, and equa­tions (7) and (8) still re­main true. The only point is that in the defin­i­tion of the $z_j$’s we had to take the $p_i$-th and $q_i$-th powers of the equa­tions in (2), and one can­not in gen­er­al take real powers of com­plex num­bers in a well-defined way, but since the left-hand sides of the ex­pres­sions in (2) are near to 1 for small de­form­a­tions of the ori­gin­al value $\mathit{z}={\mathit{z}}^0$ and since a com­plex num­ber near 1 has a well-defined log­ar­ithm near 0, there is no prob­lem. When the $p_i$ and $q_i$ are not in­teg­ral, we are no longer “filling in” the cusp by glu­ing on a sol­id tor­us, but are simply chan­ging the hy­per­bol­ic struc­ture on the ori­gin­al open to­po­lo­gic­al man­i­fold $M$, with the new hy­per­bol­ic struc­tures in gen­er­al be­ing in­com­plete. If we now define $2h$ com­plex num­bers $\mathfrak{u}=({\mathfrak{u}}_1,\dots ,{\mathfrak{u}}_h)$ and $\mathfrak{v}=({\mathfrak{v}}_1,\dots ,{\mathfrak{v}}_h)$ by $${\mathfrak{u}}_i=\sum _{j=1}^N(M_{ij}^{\prime }\log \frac{z_j}{z_j^0}+M_{ij}^{\prime \prime }\log \frac{1-z_j}{1-z_j^0}),{\mathfrak{v}}_i=\sum _{j=1}^N(L_{ij}^{\prime }\log \frac{z_j}{z_j^0}+L_{ij}^{\prime \prime }\log \frac{1-z_j}{1-z_j^0}),$$ then from the sym­plect­ic prop­er­ties of the glu­ing equa­tions it fol­lows that ([2], Lemma 4.1) $$p_i{\mathfrak{u}}_i+q_i{\mathfrak{v}}_i=2\pi i(i=1,\dots ,h),$$ and we can take $\mathfrak{u}$ as ca­non­ic­al co­ordin­ates for the above-men­tioned neigh­bor­hood $\mathfrak{U}$ of $0\in {\mathbb{C}}^h$, in which case each ${\mathfrak{v}}_i$ be­comes an odd power series in the $\mathfrak{u}$’s with lin­ear term ${\tau }_i{\mathfrak{u}}_i$ and we can write $L_i(\mathfrak{u})$ and $\mathrm{Vol}(\mathfrak{u})$ in­stead of $L_i(\kappa )$ and $\mathrm{Vol}(M_{\kappa })$. We should men­tion that $\mathfrak{u}$ and $\mathfrak{v}$ can be defined in­vari­antly, without us­ing any tri­an­gu­la­tion, as fol­lows: the de­formed hy­per­bol­ic struc­ture on $M$ cor­res­ponds to a ho­mo­morph­ism $\rho :\mathrm{\Gamma }\to {\mathrm{PSL}}_2(\mathbb{C})$ near to the in­clu­sion map, and then $\mathfrak{u}$ and $\mathfrak{v}$ are simply the log­ar­ithms of the ra­tios of the ei­gen­val­ues of the im­ages un­der $\rho$ of the me­ridi­ans and lon­git­udes, re­spect­ively.

In terms of the new co­ordin­ates, equa­tion (8b) be­comes $$\begin{array}{}\text{(9)}& \mathrm{Vol}(\mathfrak{u})=\mathrm{Vol}(M)-\frac{\pi }{2}\sum _{i=1}^h{L}_i(\mathfrak{u})+\epsilon (\mathfrak{u}),\end{array}$$ with $\epsilon (\mathfrak{u})=\mathrm{O}(\parallel \mathfrak{u}{\parallel }^4)$ as $\mathfrak{u}$ tends to 0 in ${\mathbb{C}}^h$. The­or­em 2 of [1] was the state­ment that the func­tion $\epsilon (\mathfrak{u})$ defined by (9) is har­mon­ic, and hence is the real part of a holo­morph­ic func­tion $f(\mathfrak{u})$ near 0 (uniquely de­term­ined if we fix $f(0)=0$). The­or­em 3, proved us­ing equa­tion (3), said that $\partial {\mathfrak{v}}_i/\partial {\mathfrak{u}}_j$ is sym­met­ric in $i$ and $j$, which im­plies that there is a single func­tion $\mathrm{\Phi }(\mathfrak{u})$ with $\partial \mathrm{\Phi }/\partial {\mathfrak{u}}_i=2{\mathfrak{v}}_i$ for all $i$, and that $f$ is giv­en in terms of $\mathrm{\Phi }$ by $4f=\mathrm{\Phi }-\mathfrak{u}\cdot \mathfrak{v}$ (or equi­val­ently by $-8f=(E-2)\mathrm{\Phi }$, where $E=\sum {\mathfrak{u}}_i\partial /\partial {\mathfrak{u}}_i$ is the Euler op­er­at­or), so that the volume cor­rec­tion $\epsilon (\mathfrak{u})$ in (9) is giv­en by $$\begin{array}{}\text{(10)}& \epsilon (\mathfrak{u})=\mathrm{Im}(f(\mathfrak{u})),f(\mathfrak{u}):=\frac{1}{4}{\int }_0^{\mathfrak{u}}(\sum _{i=1}^h({\mathfrak{v}}_{i}d{\mathfrak{u}}_i-{\mathfrak{u}}_{i}d{\mathfrak{v}}_i)).\end{array}$$ The func­tion $f$ is now of­ten called the Neu­mann–Za­gi­er po­ten­tial func­tion, al­though this name was used in the ori­gin­al pa­per for $\mathrm{\Phi }$ in­stead. It should per­haps also be men­tioned that sim­pler proofs of the last res­ults de­scribed could prob­ably have been ob­tained by us­ing the second rather than the first volume for­mula in (4).

There is one more im­port­ant point about volumes. An­oth­er in­sight by Thur­ston was that the volume of a hy­per­bol­ic 3-man­i­fold, which is a pos­it­ive real num­ber, is ac­tu­ally in a nat­ur­al way the ima­gin­ary part of a com­plexi­fied volume whose real part is the Chern–Si­mons in­vari­ant, an im­port­ant to­po­lo­gic­al in­vari­ant tak­ing val­ues in $\mathbb{R}/(4{\pi }^2\mathbb{Z})$ whose defin­i­tion we omit here. It was con­jec­tured in [1], and proved soon af­ter­wards by Yoshida [e2], that the above for­mu­las re­main true with the volumes re­placed by their com­plexi­fied ver­sions, the func­tions $L_i(\mathfrak{u})$ also lif­ted suit­ably from $\mathbb{R}$ to $\mathbb{C}$, and $\epsilon (\mathfrak{u})$ re­placed by $f(\mathfrak{u})$. Later, in [2], Wal­ter showed how to lift (6) to an ex­pli­cit and com­put­able ex­pres­sion for the com­plexi­fied volume of $M_{\mathit{p},\mathit{b}\mathit{q}}$ in terms of the com­plex di­log­ar­ithm.

The Bloch group of a field is an ana­logue of its mul­ti­plic­at­ive group, but with the re­la­tion $[xy]=[x]+[y]$ sat­is­fied by the log­ar­ithm func­tion re­placed by the func­tion­al equa­tion of the di­log­ar­ithm. In this sec­tion we re­call its defin­i­tion and the defin­i­tion of the “ex­ten­ded Bloch group” that was in­tro­duced by Wal­ter [4] and fur­ther de­veloped by Zick­ert and Goette [e15], [e26], and ex­plain their con­nec­tions with the volume and com­plexi­fied volume. The next sec­tion tells how these things re­late to the sym­plect­ic struc­ture. We should men­tion that parts of both sec­tions have been trans­ferred here from the arX­iv ver­sion of [e33] and also ed­ited some­what for the pur­pose of the present ex­pos­i­tion.

The di­log­ar­ithm func­tion ${\mathrm{Li}}_2(z)$, defined for $|z|<1$ as $\sum _{n=1}^{\mathrm{\infty }}{z}^n/n^2$ and then ex­ten­ded ana­lyt­ic­ally to either the cut plane $\mathbb{C}\setminus [1,\mathrm{\infty })$ or to the uni­ver­sal cov­er of ${\mathbb{P}}^1(\mathbb{C})\setminus \{0,1,\mathrm{\infty }\}$, sat­is­fies a fam­ous func­tion­al equa­tion called the five-term re­la­tion. This func­tion­al equa­tion was dis­covered re­peatedly dur­ing the 19th cen­tury and can be writ­ten in many equi­val­ent forms, each say­ing that a sum of five di­log­ar­ithm val­ues is a lin­ear com­bin­a­tion of products of simple log­ar­ithms. The func­tion ${\mathrm{Li}}_2$ is many-val­ued, but the mod­i­fied di­log­ar­ithm (5) is a single-val­ued real ana­lyt­ic func­tion from ${\mathbb{P}}^1(\mathbb{C})\setminus \{0,1,\mathrm{\infty }\}$ to $\mathbb{R}$ that ex­tends con­tinu­ously to all of ${\mathbb{P}}^1(\mathbb{C})$ and sat­is­fies “clean” ver­sions of the five-term re­la­tions with no log­ar­ithmic cor­rec­tion terms. Since $D(z)$ also sat­is­fies the two func­tion­al equa­tions $D(1-z)=-D(z)=D(1/z)$ (im­ply­ing that $D(z)=D(z^{\prime })=D(z^{\prime \prime })$ for the three shape para­met­ers of an ori­ented ideal hy­per­bol­ic tet­ra­hed­ron), this “clean” func­tion­al equa­tion still can be writ­ten in many dif­fer­ent forms, one stand­ard one be­ing $$D(x)+D(y)+D(\frac{1-x}{1-xy})+D(xy)+D(\frac{1-y}{1-xy})=0$$ for $(x,y)\ne (1,1)$ in ${\mathbb{C}}^2$. An­oth­er nice ver­sion is the cyc­lic one $\sum _{i(\mathrm{mod}5)}D(z_i)=0$ if $\{z_i{\}}_{i\in \mathbb{Z}}$ is a se­quence of com­plex num­bers sat­is­fy­ing $1-z_i=z_{i-1}{z}_{i+1}$ for all $i$ (which im­plies by a short cal­cu­la­tion that they have peri­od 5). Yet an­oth­er, with a clear in­ter­pret­a­tion in terms of 3-di­men­sion­al hy­per­bol­ic geo­metry, says that the signed sum of $D(r_i)$ is 0 if $r_1,\dots ,r_5$ are the cross-ra­tios of the five sub­sets of car­din­al­ity 4 of a set of five dis­tinct points in ${\mathbb{P}}^1(\mathbb{C})$.

The five ar­gu­ments $z_i$ of any ver­sion of the five-term re­la­tion sat­is­fy $\sum (z_i)\wedge (1-z_i)=0$, where the sum is taken in the second ex­ter­i­or power of the mul­ti­plic­at­ive group of $\mathbb{C}$. For in­stance, for the “cyc­lic ver­sion” above we have $$\sum _i(z_i)\wedge (1-z_i)=\sum _i(z_i)\wedge (z_{i-1}{z}_{i+1})=\sum _i((z_i)\wedge (z_{i-1})+(z_{i-1})\wedge (z_i))=0.$$ The Bloch group $\mathcal{B}(F)$ of an ar­bit­rary field $F$, in­tro­duced by Bloch [e1] in 1978, is mo­tiv­ated by this ob­ser­va­tion and is defined as the quo­tient of the ker­nel of the map $d:\mathbb{Z}[F]\to {\mathsf{Li}}^2(F^{\times })$ send­ing $[x]$ to $x\wedge (1-x)$ for $x\ne 0,1$ (and to 0 for $x=0,1$) by the sub­group gen­er­ated by the five-term re­la­tion of the di­log­ar­ithm. The pre­cise defin­i­tion var­ies slightly in the lit­er­at­ure be­cause of del­ic­ate 2- and 3-tor­sion is­sues arising from the par­tic­u­lar defin­i­tion of the ex­ter­i­or square (for in­stance, does one re­quire $x\wedge x=0$ for all $x$ or just $x\wedge y=-y\wedge x$?), wheth­er one re­quires $d([0])$ and $d([1])$ to van­ish or merely to be tor­sion, and the par­tic­u­lar ver­sion of the five-term re­la­tion used. We will gloss over this point for now, but will come back to it in con­nec­tion with the ex­ten­ded Bloch group.

From our point of view, the clearest mo­tiv­a­tion for the defin­i­tion of the Bloch group is the fact that the shape para­met­ers $\{z_i\}$ for any ideal tri­an­gu­la­tion $\bigcup _i\mathrm{\Delta }(z_i)$ of a com­plete hy­per­bol­ic 3-man­i­fold $M$ sat­is­fy $$\sum _i(z_i)\wedge (1-z_i)=0.$$ (This is a con­sequence of the sym­plect­ic nature of the NZ re­la­tions, as we will ex­plain in more de­tail in the next sec­tion.) Thus to any such tri­an­gu­la­tion we can as­so­ci­ate a class $\sum _i[z_i]$ in the Bloch group. But this class is in fact in­de­pend­ent of the tri­an­gu­la­tion, since (mod­ulo some tech­nic­al points con­cern­ing the fact that the shapes can de­gen­er­ate to 0 or 1 un­der 2-3 Pach­ner moves) any two tri­an­gu­la­tions are linked by a series of “2-3 Pach­ner moves” in which two tet­ra­hedra shar­ing a com­mon face are re­placed by the three tet­ra­hedra defined by their two non­shared and two of their three shared ver­tices, and the (signed) sum of the shape para­met­ers of these five tet­ra­hedra is pre­cisely the five-term re­la­tion and does not af­fect the class of $\sum _i[z_i]$ in the Bloch group. Thus one has a class $[M]\in \mathcal{B}(\mathbb{C})$. Moreover, from the very defin­i­tion of the Bloch group it fol­lows that the func­tion $D$ ex­tends to a lin­ear map from $\mathcal{B}(\mathbb{C})$ to $\mathbb{R}$, and from the dis­cus­sion in the last sec­tion we see that the value of $D$ on the class $[M]$ is equal to the volume of $M$. Al­though we will not use it, we men­tion that by a res­ult of Suslin the Bloch group $\mathcal{B}(F)$ of any field $F$ is iso­morph­ic up to tor­sion to the al­geb­ra­ic $K$-group $K_3(F)$, with $D$ cor­res­pond­ing to the Borel reg­u­lat­or map from $K_3(\mathbb{C})$ to $\mathbb{R}$ in the case $F=\mathbb{C}$.

On the oth­er hand, as de­scribed at the end of the last sec­tion, the hy­per­bol­ic volume should ac­tu­ally be seen as the ima­gin­ary part of a com­plexi­fied volume tak­ing val­ues in $\mathbb{C}/4{\pi }^2\mathbb{Z}$, so we would like to re­place the func­tion $D(z)$ by some com­plex-val­ued ver­sion of the di­log­ar­ithm which, even though it may be many-val­ued at in­di­vidu­al ar­gu­ments $z$, be­comes one-val­ued mod­ulo $4{\pi }^2$ if we take a lin­ear com­bin­a­tion of its val­ues with ar­gu­ments be­long­ing to the Bloch group. This is the idea be­hind the pas­sage from the ori­gin­al Bloch group to the ex­ten­ded one. The first ob­ser­va­tion (see [e9]) is that the func­tion $L(v):={\mathrm{Li}}_2(1-e^v)$ has the de­riv­at­ive $v/(e^{-v}-1)$, which is mero­morph­ic and has residues in $2\pi i\mathbb{Z}$, so that $L$ it­self lifts to a well-defined func­tion from $\mathbb{C}\setminus 2\pi i\mathbb{Z}$ to $\mathbb{C}/4{\pi }^2\mathbb{Z}$ and sat­is­fies the func­tion­al equa­tion $$L(v+2\pi in)=L(v)-2\pi in\log (1-e^v)\text{for }n\in \mathbb{Z}.$$ We now in­tro­duce the com­plex 1-man­i­fold $$\hat{\mathbb{C}}=\{(u,v)\in {\mathbb{C}}^2\mid e^u+e^v=1\}.$$ This is an abeli­an cov­er of ${\mathbb{C}}^{\times }\setminus \{0,1\}$ via $z=e^u=1-e^v$, with Galois group iso­morph­ic to ${\mathbb{Z}}^2$. The ex­ten­ded Bloch group $\hat{\mathcal{B}}(\mathbb{C})$ as defined in [e15] or [e26] is the ker­nel of the map $\hat{d}:\mathbb{Z}[\hat{\mathbb{C}}]\to {\mathsf{Li}}^2(\mathbb{C})$, where ${\mathsf{Li}}^2(\mathbb{C})$ is defined by re­quir­ing only $x\wedge y+y\wedge x=0$ (rather than $x\wedge x=0$, which is stronger by 2-tor­sion) and where $\hat{d}$ maps $[u,v]:=[(u,v)]\in \mathbb{Z}[\hat{\mathbb{C}}]$ to $u\wedge v$, di­vided by an ap­pro­pri­ate lif­ted ver­sion of the five-term re­la­tion, namely, the $\mathbb{Z}$-span of the set of ele­ments $\sum _{j=1}^5(-1{)}^j[u_j,v_j]$ of $\mathbb{Z}(\hat{\mathbb{C}})$ sat­is­fy­ing $$(u_2,u_4)=(u_1+u_3,u_3+u_5)\text{and}(v_1,v_3,v_5)=(u_5+v_2,v_2+v_4,u_1+v_4).$$ There is an ex­ten­ded reg­u­lat­or map from $\hat{\mathcal{B}}(\mathbb{C})$ to $\mathbb{C}/4{\pi }^2\mathbb{Z}$ giv­en by map­ping $\sum [u_j,v_j]$ to $\sum \mathcal{L}(u_j,v_j)$, where $$\mathcal{L}(u,v)=L(v)+\frac{1}{2}uv-\frac{{\pi }^2}{6},$$ which one can check van­ishes mod­ulo $4{\pi }^2$ on the lif­ted five-term re­la­tion. One can also define $\hat{\mathcal{B}}(F)$ for any sub­field $F$ of $\mathbb{C}$, such as an em­bed­ded num­ber field, by re­pla­cing $\hat{\mathbb{C}}$ by the sub­set $\hat{F}$ con­sist­ing of pairs $(u,v)$ with $e^u=1-e^v\in F$.

As a fi­nal re­mark, one can won­der to what ex­tent study­ing just hy­per­bol­ic 3-man­i­folds lets one un­der­stand the full Bloch group of $\bar{\mathbb{Q}}$. For in­stance, does every ele­ment of $\mathcal{B}(\bar{\mathbb{Q}})$ oc­cur as a ra­tion­al lin­ear com­bin­a­tion of the Bloch group in­vari­ants of some hy­per­bol­ic 3-man­i­folds? Even more ba­sic­ally, does every num­ber field with at least one non­real em­bed­ding oc­cur as the trace field of some hy­per­bol­ic 3-man­i­fold? The lat­ter ques­tion was posed ex­pli­citly by Wal­ter in [3].

In ret­ro­spect, the sym­plect­ic prop­er­ties as de­scribed in equa­tion (3) and the fol­low­ing text, and their re­fine­ment from $\mathbb{Q}$ to $\mathbb{Z}$ as giv­en in the fol­low-up pa­per [2], turned out to be the most im­port­ant as­pects of these pa­pers. They are re­spons­ible both for the ex­ist­ence of the po­ten­tial func­tion and for all of the ap­plic­a­tions to quant­iz­a­tion that we will de­scribe in the next sec­tion, as well as many of the con­nec­tions to num­ber the­ory de­scribed in Sec­tion 6.

Define an $N\times 2N$ mat­rix $H=(AB)$ whose rows form a $\mathbb{Z}$-basis for the lat­tice spanned by the edge equa­tions (1) to­geth­er with one “peri­pher­al” equa­tion (a coprime lin­ear com­bin­a­tion of the me­ridi­an and lon­git­ude equa­tions in (2)) at each cusp. Then the above cited res­ults in [1] im­ply that $A{B}^t$ is sym­met­ric and that $H$ has rank $N$, mean­ing that its $2N$ columns gen­er­ate ${\mathbb{Q}}^N$. To­geth­er, these two state­ments are equi­val­ent to say­ing that $H$ can be ex­ten­ded to a $2N\times 2N$ mat­rix $$(\begin{array}{cc}A& B\\ C& D\end{array})$$ in ${\mathrm{Sp}}_{2N}(\mathbb{Q})$, mean­ing that $$(\begin{array}{cc}A& B\\ C& D\end{array}{)}^{-1}=(\begin{array}{cc}\phantom{-}{D}^t& -B^t\\ -C^t& \phantom{-}{A}^t\end{array}).$$

But in fact the $2N$ columns of $H$ span the lat­tice ${\mathbb{Z}}^N$, which is equi­val­ent to say­ing that $H$ can be com­pleted to a $2N\times 2N$ sym­plect­ic mat­rix over $\mathbb{Z}$. (We will call such a mat­rix half-sym­plect­ic.) This fol­lows from the chain com­plex defined by Wal­ter in [2]. Ex­pli­citly, for any sim­plex $\mathrm{\Delta }$, let $J_{\mathrm{\Delta }}$ be the abeli­an group gen­er­ated by $e_1,e_2,e_3$ (cor­res­pond­ing to the pairs of op­pos­ite edges) sub­ject to the re­la­tion $e_1+e_2+e_3=0$. This is a free abeli­an group of rank 2, with a ca­non­ic­al nonsin­gu­lar, skew-sym­met­ric bi­lin­ear form giv­en by ([2], Sec­tion 4) $$\begin{array}{}\text{(11)}& ⟨e_1,e_2⟩=⟨e_2,e_3⟩=⟨e_3,e_1⟩=-⟨e_2,e_1⟩=-⟨e_3,e_2⟩=-⟨e_1,e_3⟩=1.\end{array}$$ The Neu­mann chain com­plex as­so­ci­ated to an ideal tri­an­gu­la­tion is then defined by $$\begin{array}{}\text{(12)}& 0⟶C_0\stackrel{\alpha }{⟶}{C}_1\stackrel{\beta }{⟶}J\stackrel{{\beta }^{\ast }}{⟶}{C}_1\stackrel{{\alpha }^{\ast }}{⟶}{C}_0⟶0.\end{array}$$ Here $C_0$ and $C_1$ are the free abeli­an groups on the un­ori­ented 0- and 1-sim­plices (cusps and edges), re­spect­ively, and $$J=\underset{i=1}{\overset{N}{⨁}}{J}_{{\mathrm{\Delta }}_i}$$ (sum over the 3-sim­plices or tet­ra­hedra), while $\alpha$ maps any cusp to the sum of its in­cid­ent edges, the $J_{\mathrm{\Delta }}$-com­pon­ent of $\beta$ of any edge is the sum of the edges of $\mathrm{\Delta }$ that are iden­ti­fied with it, and ${\alpha }^{\ast }$ and ${\beta }^{\ast }$ are the du­als of $\alpha$ and $\mathcal{B}$ with re­spect to the ob­vi­ous scal­ar products on $C_i$ and the sym­plect­ic form on $J$. Wal­ter shows ([2], The­or­em 4.1) that the se­quence (12) is a chain com­plex and, at least after tensor­ing with $\mathbb{Z}[\frac{1}{2}]$, is ex­act ex­cept in the middle, where the ho­mo­logy is the sum of $h$ rank-2 mod­ules iso­morph­ic to $H_1(T_i^2)$ $(i=1,\dots ,h)$. Note that the map $\beta$ is giv­en pre­cisely by the mat­rix $R=(R^{\prime }{R}^{\prime \prime })$ as defined in (1) if we choose the ob­vi­ous basis for $C_1$ and the basis of $J$ giv­en by choos­ing the basis $(e_1,e_2)$ for every $J_{{\mathrm{\Delta }}_j}$. The rest of the proof that $H$ is half-sym­plect­ic fol­lows eas­ily from the the­or­em just quoted and will be left to the read­er.

We make two re­marks about this. The first is that both the con­struc­tion of the chain com­plex and the state­ment about its ho­mo­logy were done in [2] also for 3-man­i­folds with bound­ary com­pon­ents of ar­bit­rary genus (so the ver­tices of $C_0$ need not be cusps), and of course also do not re­quire any hy­per­bol­ic struc­ture. The oth­er is that the glu­ing equa­tions of [1] and the sym­plect­ic res­ults of [2] were ex­ten­ded to ar­bit­rary ${\mathrm{PGL}}_n$-rep­res­ent­a­tions in [e29].

Half-sym­plect­ic matrices oc­cur in oth­er con­texts, e.g., in con­nec­tion with Nahm’s con­jec­ture on the mod­u­lar­ity of cer­tain $q$-hy­per­geo­met­ric series, and also lead to a new de­scrip­tion of the Bloch group. Both top­ics will be dis­cussed in more de­tail in Sec­tion 6.

Per­haps the most far-reach­ing con­sequences of Wal­ter’s work on the com­bin­at­or­ics of 3-di­men­sion­al tri­an­gu­la­tions have been the ap­plic­a­tions of the sym­plect­ic struc­ture to quant­iz­a­tion.

Re­call the defin­i­tion of $J_{\mathrm{\Delta }}$ for a single tet­ra­hed­ron $\mathrm{\Delta }$ as the abeli­an group $$⟨e_1,e_2,e_3\mid e_1+e_2+e_3=0⟩$$ with the sym­plect­ic struc­ture (11). This sym­plect­ic struc­ture on each space $J_{\mathrm{\Delta }}{\otimes }_{\mathbb{Z}}\mathbb{Q}$ for any ideal tet­ra­hed­ron $\mathrm{\Delta }$ leads to an in­teg­ral Lag­rangi­an sub­space of the 10-di­men­sion­al sym­plect­ic space $$\underset{j=1}{\overset{5}{⨁}}{J}_{{\mathrm{\Delta }}_j}{\otimes }_{\mathbb{Z}}\mathbb{Q}$$ as­so­ci­ated to five tet­ra­hedra ${\mathrm{\Delta }}_1,\dots ,{\mathrm{\Delta }}_5$ that par­ti­cip­ate in a 2-3 Pach­ner move. Roughly speak­ing, the Lag­rangi­an sub­space re­cords the lin­ear re­la­tions among the angles of the five tet­ra­hedra, where the signed sum of the angles around each in­teri­or edge of the Pach­ner move is zero.

The quant­iz­a­tion of this Lag­rangi­an sub­space has ap­peared nu­mer­ous times in the math­em­at­ics and phys­ics lit­er­at­ure, un­der dif­fer­ent names, and has led to in­ter­est­ing quantum in­vari­ants in di­men­sions 2, 3 and 4. We briefly dis­cuss this now. In di­men­sion 2, Kashaev and in­de­pend­ently Fo­ck–Gon­char­ov [e8], [e17], [e13] used the above NZ-sym­plect­ic struc­ture to study the change of co­ordin­ates of ideally tri­an­gu­lated sur­faces un­der a 2-2 Pach­ner move. They found that the cor­res­pond­ing iso­morph­ism of com­mut­at­ive al­geb­ras can be de­scribed in terms of cluster al­geb­ras, lead­ing to two dual sets of co­ordin­ates (the so-called $\mathcal{X}$-co­ordin­ates and the $\mathcal{A}$-co­ordin­ates) whose quant­iz­a­tion leads to a rep­res­ent­a­tion of the so-called Ptolemy group­oid, and in par­tic­u­lar of the map­ping class group of a punc­tured sur­face, and also of braid groups. These rep­res­ent­a­tions are al­ways in­fin­ite-di­men­sion­al (be­cause there are no fi­nite square matrices $A$ and $B$ sat­is­fy­ing the re­la­tion $AB-BA=I$), the Hil­bert spaces are typ­ic­ally $L^2({\mathbb{R}}^n)$ for some $n$, and the cor­res­pond­ing the­ory is usu­ally known as quantum Teichmüller the­ory.

Go­ing one di­men­sion high­er, the $\mathcal{X}$-co­ordin­ates of a 3-di­men­sion­al ideal tri­an­gu­la­tion are noth­ing but the shapes of the ideal tet­ra­hedra, where­as the $\mathcal{A}$-co­ordin­ates are the Ptolemy vari­ables of the ideal tet­ra­hedra. The lat­ter are as­sign­ments of nonzero com­plex num­bers to the edges of the ideal tri­an­gu­la­tion (where iden­ti­fied edges are giv­en the same vari­able) that sat­is­fy a sys­tem of quad­rat­ic equa­tions: a (suit­ably) signed sum $ab+cd=ef$, where $(a,b)$, $(b,d)$ and $(e,f)$ are the Ptolemy vari­ables of the three pairs of op­pos­ite edges. It turns out that the NZ glu­ing equa­tions for shapes are equi­val­ent to the Ptolemy equa­tions (see for in­stance [e25]), and this is not only the­or­et­ic­ally in­ter­est­ing, but prac­tic­ally, too. The quant­iz­a­tion of the shape and Ptolemy vari­ables of an ideal tri­an­gu­la­tion uses two in­gredi­ents, the kin­emat­ic­al ker­nel of Kashaev [e30] and a spe­cial func­tion, the Fad­deev quantum di­log­ar­ithm that sat­is­fies an in­teg­ral pentagon iden­tity. Ac­cord­ing to Kashaev, the kin­emat­ic­al ker­nel is noth­ing but the quant­iz­a­tion of the NZ Lag­rangi­an men­tioned above. The out­come of this quant­iz­a­tion is the ex­ist­ence of to­po­lo­gic­al in­vari­ants of ideally tri­an­gu­lated 3-man­i­folds, the in­vari­ants be­ing ana­lyt­ic func­tions in a cut place ${\mathbb{C}}^{\prime }=\mathbb{C}\setminus (-\mathrm{\infty },0]$, ex­pressed in terms of fi­nite-di­men­sion­al state in­teg­rals whose in­teg­rand is of­ten de­term­ined by the com­bin­at­or­i­al data of an ideal tri­an­gu­la­tion, namely its Neu­mann–Za­gi­er matrices. This con­struc­tion, which is of­ten known as quantum hy­per­bol­ic geo­metry, has been ax­io­mat­ized by Kashaev, and uses as in­put the com­bin­at­or­i­al data of an ideal tri­an­gu­la­tion to­geth­er with a self-dual loc­ally com­pact abeli­an group with fixed Gaus­si­an, Four­i­er ker­nel and quantum di­log­ar­ithm. This then leads to fur­ther ana­lyt­ic in­vari­ants of 3-man­i­folds, two ex­amples of which are the Kashaev–Luo–Vartan­ov in­vari­ants [e28] and the mero­morph­ic 3D-in­dex [e32], for which the LCA groups are $\mathbb{R}\times \mathbb{R}$ and $S^1\times \mathbb{Z}$, re­spect­ively. It is worth not­ing that the An­der­sen–Kashaev state in­teg­rals are con­jec­tured to be the par­ti­tion func­tion of com­plex Chern–Si­mons the­ory (i.e., Chern–Si­mons the­ory with com­plex gauge group). The lat­ter is not known to sat­is­fy the cut-and-paste ar­gu­ments that the $\mathrm{SU}(2)$ Chern–Si­mons the­ory does, and as a res­ult, one does not have an a pri­ori defin­i­tion of com­plex Chern–Si­mons the­ory oth­er than the state in­teg­rals, nor a clear reas­on why the in­fin­ite-di­men­sion­al path in­teg­ral loc­al­izes to a fi­nite-di­men­sion­al one.

Fi­nally, go­ing yet one di­men­sion high­er, the five ideal tet­ra­hedra that par­ti­cip­ate in a 2-3 Pach­ner move form the bound­ary of a single 4-di­men­sion­al sim­plex, a penta­choron. (Ex­cuse our Greek.) This gives a 4-di­men­sion­al in­ter­pret­a­tion of the NZ-struc­ture and of the kin­emat­ic­al ker­nel, and us­ing a com­plex root of unity, Kashaev was able to give a tensor in­vari­ant un­der 4-di­men­sion­al Pach­ner moves and thus con­struct cor­res­pond­ing to­po­lo­gic­al in­vari­ants of closed, tri­an­gu­lated 4-man­i­folds at roots of unity [e30]. This con­cludes our dis­cus­sion of the kin­emat­ic­al ker­nel in di­men­sions 2, 3 and 4.

In a dif­fer­ent dir­ec­tion, math­em­at­ic­al phys­i­cists, us­ing cor­res­pond­ence prin­ciples among su­per­sym­met­ric the­or­ies, have came up with un­ex­pec­ted con­struc­tions of vari­ous col­lec­tions of $q$-series with in­teger coef­fi­cients as­so­ci­ated to 3-man­i­folds. Per­haps the most re­mark­able of these is the 3D-in­dex of Dimofte, Gai­otto and Gukov [e23], [e22], where the $q$-series in ques­tion, which in this case are in­dexed by pairs of in­tegers, were defined ex­pli­citly in terms of the NZ-matrices of a suit­able ideal tri­an­gu­la­tion, with their coef­fi­cients count­ing the num­ber of BPS states of a su­per­sym­met­ric the­ory. This DGG 3D-in­dex was sub­sequently shown [e27] to be a to­po­lo­gic­al in­vari­ant of cusped hy­per­bol­ic 3-man­i­folds, and was also ex­ten­ded to a mero­morph­ic func­tion of two vari­ables (in case the bound­ary of the 3-man­i­fold is a single tor­us) whose Laurent coef­fi­cients are the DGG in­dex [e32].

A quite dif­fer­ent place where the NZ equa­tions ap­pear in quantum to­po­logy is in con­nec­tion with the Kashaev in­vari­ant and Kashaev’s fam­ous Volume Con­jec­ture. The Kashaev in­vari­ant $⟨K{⟩}_n$ is a com­put­able al­geb­ra­ic num­ber that was defined for any knot $K$ and any pos­it­ive in­teger $n$ by Kashaev [e5] in 1995 us­ing ideas of quantum to­po­logy sim­il­ar to those dis­cussed above, and of which an al­tern­at­ive defin­i­tion in terms of the so-called colored Jones poly­no­mi­al was later found by H. Murakami and J. Murakami. The Volume Con­jec­ture [e6] says that the log­ar­ithm of $|⟨K{⟩}_n|$ is asymp­tot­ic­ally equal to $n/(2\pi )$ times the hy­per­bol­ic volume of the knot com­ple­ment $M=S^3\setminus K$ whenev­er $M$ is hy­per­bol­ic, a very sur­pris­ing con­nec­tion between hy­per­bol­ic geo­metry and 3-di­men­sion­al quantum to­po­logy that has giv­en rise to a great deal of sub­sequent re­search and has been re­fined in many ways and by sev­er­al au­thors in con­nec­tion with com­plex Chern–Si­mons the­ory. In par­tic­u­lar, one has the con­jec­tur­al sharpen­ing [e16], [e18] $$\begin{array}{}\text{(13)}& ⟨K{⟩}_n\sim n^{3/2}{e}^{{\mathrm{V}}_{\mathbb{C}}(K)n/(2\pi i)}{\mathrm{\Phi }}^K(\frac{2\pi i}{n})\end{array}$$ to all or­ders in $1/n$ as $n\to \mathrm{\infty }$, where ${\mathrm{\Phi }}^K(h)$ is a power series in $h$ with al­geb­ra­ic coef­fi­cients that can be com­puted to any or­der in any ex­pli­cit ex­ample, e.g., $$\begin{array}{}\text{(14)}& {\mathrm{\Phi }}^{4_1}(h)=\frac{1}{\sqrt[4]{3}}(1+\frac{11}{72\sqrt{-3}}h+\frac{697}{2(72\sqrt{-3}{)}^2}{h}^2+\frac{724351}{30(72\sqrt{-3}{)}^3}{h}^3+\cdots )\end{array}$$ for the $4_1$ (fig­ure-8) knot. In [e20], an ex­pli­cit can­did­ate for this power series is con­struc­ted for any knot as a form­al Gaus­si­an in­teg­ral whose in­teg­rand is defined in terms of the NZ data of an ideal tri­an­gu­la­tion of $M$. It is not yet known bey­ond the lead­ing term that the series con­struc­ted there is a to­po­lo­gic­al in­vari­ant (i.e., in­de­pend­ent of the choice of tri­an­gu­la­tion), al­though this would of course fol­low from the con­jec­ture that the asymp­tot­ic for­mula (13) holds with this series. In a fol­low-up pa­per [e31], the con­struc­tion was ex­ten­ded, still us­ing NZ data in an es­sen­tial way, to give an ex­pli­citly com­put­able power series ${\mathrm{\Phi }}_{\alpha }^K(h)$ for any $\alpha \in \mathbb{Q}/\mathbb{Z}$, with ${\mathrm{\Phi }}_0^K={\mathrm{\Phi }}^K$, that is ex­pec­ted to be the power series pre­dicted by the quantum mod­u­lar­ity con­jec­ture for knots that we will dis­cuss in the next sec­tion.

Fi­nally, it is worth not­ing that the NZ-equa­tions and their sym­plect­ic prop­er­ties lead to an ex­pli­cit quant­iz­a­tion of the shape vari­ables, where one re­places each $z$, $z^{\prime }$ and $z^{\prime \prime }$ by op­er­at­ors that suit­ably com­mute. This was car­ried out by Dimofte [e21], who defined a quant­ized ver­sion of the glu­ing equa­tions, a so-called quantum curve, which is ex­pec­ted to an­ni­hil­ate the par­ti­tion func­tion of com­plex Chern–Si­mons the­ory and to be ul­ti­mately re­lated to the asymp­tot­ics of quantum in­vari­ants.

The pa­per [1] and its se­quel [2] sug­ges­ted or led to sev­er­al in­ter­est­ing de­vel­op­ments in pure num­ber the­ory as well as in to­po­logy. In this fi­nal sec­tion we de­scribe of a few of these.

An im­port­ant sub­class of hy­per­bol­ic man­i­folds $M={\mathbb{H}}^3/\mathrm{\Gamma }$ are the arith­met­ic ones, where $\mathrm{\Gamma }$ is either the Bi­an­chi group ${\mathrm{SL}}_2({\mathcal{O}}_F)$ for some ima­gin­ary quad­rat­ic field $F$ or more gen­er­ally a group of units in a qua­ternion al­gebra over a num­ber field $F$ of high­er de­gree $n=r_1+2$ hav­ing only one com­plex em­bed­ding up to com­plex con­jug­a­tion. In both cases, clas­sic­al res­ults (proved in the first case by Hum­bert already in 1919) say that the volume of  $M$ is a simple mul­tiple (a power of $\pi$ times the square-root of the dis­crim­in­ant) of the value at $s=2$ of the Dede­kind zeta func­tion ${\zeta }_F(s)$ of the field $F$. An im­me­di­ate con­sequence of this and of the volume for­mu­las dis­cussed in Sec­tion 2 is that this zeta value is a mul­tiple of a lin­ear com­bin­a­tion of val­ues of the Bloch–Wign­er di­log­ar­ithm at al­geb­ra­ic ar­gu­ments. This con­sequence was ob­served in [e3] and was also gen­er­al­ized there to the value of ${\zeta }_F(2)$ for ar­bit­rary num­ber fields $F$, with $[F:\mathbb{Q}]=r_1+2r_2$ for any value of $r_2\ge 1$. (If $r_2=0$, then the well-known Klin­gen–Siegel the­or­em as­serts that ${\zeta }_F(2)$ is a ra­tion­al mul­tiple of ${\pi }^{2r_1}\sqrt{D_F}$.) Now the group ${\mathrm{SL}}_2({\mathcal{O}}_F)$ acts as a dis­crete group of iso­met­ries of $({\mathbb{H}}^2{)}^{r_1}\times ({\mathbb{H}}^3{)}^{r_2}$ with a quo­tient of fi­nite volume, and there are also qua­ternion­ic groups $\mathrm{\Gamma }$ over $F$ that act freely and dis­cretely on $({\mathbb{H}}^3{)}^{r_2}$, the volume of the quo­tient in both cases be­ing an ele­ment­ary mul­tiple of ${\zeta }_F(2)$. This gives a “poly-3-hy­per­bol­ic” man­i­fold $M=({\mathbb{H}}^3{)}^{r_2}/\mathrm{\Gamma }$ with volume pro­por­tion­al to ${\zeta }_F(2)$. A rather amus­ing lemma says that any such man­i­fold has a de­com­pos­i­tion (dis­joint ex­cept for the bound­ar­ies) in­to fi­nitely many $r_2$-fold products of hy­per­bol­ic tet­ra­hedra, and it fol­lows that ${\zeta }_F(2)$ for any num­ber field has an ex­pres­sion as a lin­ear com­bin­a­tion of $r_2$-fold products of val­ues of $D(z)$ at al­geb­ra­ic ar­gu­ments, gen­er­al­iz­ing the Klin­gen–Siegel the­or­em in an un­ex­pec­ted way.

The con­nec­tion with 3-di­men­sion­al hy­per­bol­ic geo­metry ap­plies only to the val­ues of Dede­kind zeta func­tions at $s=2$, but sug­ges­ted that there might be sim­il­ar state­ments for ${\zeta }_F(m)$ with $m>2$ in terms of the $m$-th poly­log­ar­ithm func­tion ${\mathrm{Li}}_m(z)$. Ex­tens­ive nu­mer­ic­al ex­per­i­ments led to a con­crete con­jec­ture say­ing that this is the case and also to a defin­i­tion (ori­gin­ally highly spec­u­lat­ive, but now sup­por­ted by more the­ory) of “high­er Bloch groups” ${\mathcal{B}}_m(F)$ that should be iso­morph­ic after tensor­ing with $\mathbb{Q}$ to the high­er al­geb­ra­ic $K$-groups $K_{2m-1}(F)$ and should ex­press the Borel reg­u­lat­or in terms of poly­log­ar­ithms. (For a sur­vey, see [e9].) This con­jec­ture, now over 30 years old, has been stud­ied ex­tens­ively by Beil­in­son, De­ligne, de Jeu, Gon­char­ov, Ruden­ko and oth­ers, with the cases $m=3$ and $m=4$ now be­ing es­sen­tially settled.

As already men­tioned in the last sec­tion, the ana­lys­is of the Kashaev in­vari­ant and the mod­u­lar gen­er­al­iz­a­tion of the Volume Con­jec­ture dis­cussed be­low led to the defin­i­tion of cer­tain power series ${\mathrm{\Phi }}_{\alpha }^K(h)$ as­so­ci­ated to a hy­per­bol­ic knot and a num­ber $\alpha \in \mathbb{Q}/\mathbb{Z}$ that can be com­puted nu­mer­ic­ally in any giv­en case. Ex­tens­ive nu­mer­ic­al com­pu­ta­tions for simple knots and simple ra­tion­al num­bers $\alpha$ sug­ges­ted that this power series not only has al­geb­ra­ic coef­fi­cients, but that (up to a root of a unity and the square-root of a num­ber in the trace field $F_K$ of the knot in­de­pend­ent of $\alpha$) its $n$-th power be­longs to $F_{K,n}[[h]]$, where $n$ is the de­nom­in­at­or of $\alpha$ and $F_{K,n}=F_K(e^{2\pi i\alpha })$ the $n$-th cyc­lo­tom­ic ex­ten­sion of $F_K$. Equi­val­ently, ${\mathrm{\Phi }}_{\alpha }^K(h)$ it­self is the product of a power series in $F_{K,n}[[h]]$ with the $n$-th root of an ele­ment of $F_{K,n}^{\times }$. Moreover, in each case stud­ied the lat­ter factor turned out to be the $n$-th root of a unit, and not just a nonzero num­ber, of $F_{K,n}$, and the “sis­ter knots” (like $5_2$ and the $(-2,3,7)$-pret­zel knot) hav­ing the same Bloch group class were the same for both knots, even though the rest of the power series were com­pletely dif­fer­ent. This led us, to­geth­er with Frank Calegari, to con­jec­ture and later to prove [e35] that there was a ca­non­ic­al class of ele­ments in cyc­lo­tom­ic ex­ten­sions of ar­bit­rary num­ber fields as­so­ci­ated to ele­ments of their Bloch groups, wheth­er or not the fields arise from to­po­logy. Ex­pli­citly, to any num­ber field $F$ and any ele­ment $\xi$ of the Bloch group of $F$ one can as­so­ci­ate ca­non­ic­ally defined ele­ments of $U(F_n)/U(F_n{)}^n$ for every $n$, where $U(F_n)$ de­notes the group of units (more pre­cisely, of $S$-units for some $S$ de­pend­ing on $\xi$ but in­de­pend­ent of $n$) of the $n$-th cyc­lo­tom­ic ex­ten­sion $F_n$ of $F$. Ac­tu­ally, two quite dif­fer­ent con­struc­tions were giv­en, one in terms of an ele­ment of $\mathcal{B}(F)$ and one in terms of an ele­ment in $K_3(F)$, and work in pro­gress an­nounced in [e35] sug­gests that there will be a gen­er­al­iz­a­tion to ${\mathcal{B}}_m(F)$ and $K_{2m-1}(F)$ for any $m>2$. The con­struc­tion in terms of the Bloch group is quite simple, al­though the proof that it gives units and is in­de­pend­ent (up to $n$-th powers) of all choices is long: if $\xi$ is rep­res­en­ted by $\sum [z_i]\in \mathbb{Z}[F]$, then the num­ber $\prod D_{\zeta }(z_i^{1/n})$ is the product of an $n$-th power in $F_n$ with an $S$-unit, and this is the unit we are look­ing for. Here $\zeta$ is a prim­it­ive $n$-th root of unity and $$D_{\zeta }(x)=\prod _{k=1}^{n-1}(1-{\zeta }^{k}x{)}^k$$ is the “cyc­lic quantum di­log­ar­ithm” func­tion, which by a res­ult of Kashaev, Mangaz­eev and Strogan­ov sat­is­fies an ana­logue of the five-term re­la­tion of the clas­sic­al di­log­ar­ithm.

An un­ex­pec­ted con­sequence of the work on units just de­scribed was a proof of one dir­ec­tion of a con­jec­ture by the math­em­at­ic­al phys­i­cist Wern­er Nahm that had pre­dicted an ex­tremely sur­pris­ing con­nec­tion between the Bloch group and the mod­u­lar­ity of cer­tain $q$-hy­per­geo­met­ric series. The simplest case of such a “Nahm sum” is the in­fin­ite series $$\begin{array}{}\text{(15)}& F_{a,b,c}(q)=\sum _{n=0}^{\mathrm{\infty }}\frac{q^{\frac{1}{2}a{n}^2+bn+c}}{(q{)}_n}(a,b,c\in \mathbb{Q},a>0),\end{array}$$ where $$(q{)}_n=(1-q)(1-q^2)\cdots (1-q^n)$$ is the so-called quantum factori­al. This func­tion is known to be mod­u­lar (in $\tau$, where $q=e^{2\pi i\tau }$) when $(a,b,c)$ is $(2,0,-\frac{1}{60})$ or $(2,1,\frac{11}{60})$ by the fam­ous Ro­gers–Ramanu­jan iden­tit­ies and in a hand­ful of oth­er cases by clas­sic­al res­ults of Euler, Gauss and oth­ers. It is very rare that a $q$-hy­per­geo­met­ric series (mean­ing an in­fin­ite sum whose ad­ja­cent terms dif­fer by fixed ra­tion­al func­tions of $q$ and $q^n$) is at the same time a mod­u­lar func­tion, and in fact for the spe­cial series (15) this hap­pens only for sev­en triples $(a,b,c)$, as pre­dicted by Nahm’s con­jec­ture and proved in [e14]. Nahm raised the gen­er­al ques­tion when a (mul­ti­di­men­sion­al) $q$-hy­per­geo­met­ric series can be mod­u­lar and, mo­tiv­ated by ex­amples com­ing from char­ac­ters of ver­tex op­er­at­or al­geb­ras, dis­covered a pos­sible an­swer in terms of the Bloch group. Con­cretely, he gen­er­al­ized (16) to $$\begin{array}{}\text{(16)}& F_{a,b,c}(q)=\sum _{n_1,\dots ,n_N\ge 0}\frac{q^{\frac{1}{2}{n}^{t}an+b^{t}n+c}}{(q{)}_{n_1}\cdots (q{)}_{n_N}}\end{array}$$ for any $N\ge 1$, where $a$ is now a pos­it­ive def­in­ite sym­met­ric $N\times N$ mat­rix with ra­tion­al coef­fi­cients, $b$ a vec­tor in ${\mathbb{Q}}^N$, and $c$ a ra­tion­al num­ber. There is still no com­plete “if and only if” con­jec­ture pre­dict­ing ex­actly when these Nahm sums are mod­u­lar func­tions, but Nahm gave a pre­cise con­jec­ture for a ne­ces­sary con­di­tion and a par­tial con­jec­ture for the suf­fi­ciency. It is the first part that was proved in [e35] while the cor­rect for­mu­la­tion and proof of the con­verse dir­ec­tion is still an act­ive re­search sub­ject.

The mod­u­lar­ity cri­terion that Nahm found de­pended on his ob­ser­va­tion that for any solu­tion $(z_1,\dots ,z_N)$ of the sys­tem of equa­tions $$\begin{array}{}\text{(17)}& 1-z_i=\prod _{j=1}^N{z}_j^{a_{ij}}(i=1,\dots ,N)\end{array}$$ the ele­ment $[z_1]+\cdots +[z_N]$ of $\mathbb{Z}[\mathbb{C}]$ be­longs to $\mathcal{B}(\mathbb{C})$. This is a dir­ect con­sequence of the sym­metry of $a$, be­cause $$\sum _i(z_i)\wedge (1-z_i)=\sum _{i,j}{a}_{ij}(z_i)\wedge (z_j)=0.$$ (This is the same ar­gu­ment as was used in Sec­tion 4 for the cor­res­pond­ing state­ment for the Neu­mann–Za­gi­er equa­tions, and ap­plies more gen­er­ally to all half-sym­plect­ic matrices, as dis­cussed be­low.) The “only if” dir­ec­tion of Nahm’s con­jec­ture then says that $F_{a,b,c}(q)$ can be mod­u­lar only if this ele­ment of the Bloch group van­ishes for the unique solu­tion of the Nahm equa­tion hav­ing all $z_i\in (0,1)$. The proof, giv­en in [e35] uses both the res­ults there about the units com­ing from Bloch ele­ments as de­scribed above and an asymp­tot­ic ana­lys­is of Nahm sums near roots of unity pub­lished sep­ar­ately by the two of us.

At the end of Sec­tion 4, we saw how the NZ equa­tions lead to a “half-sym­plect­ic mat­rix,” mean­ing the up­per half $H=(AB)$ of a $2N\times 2N$ sym­plect­ic mat­rix $$(\begin{array}{cc}A& B\\ C& D\end{array})$$ over $\mathbb{Z}$. To any such mat­rix we as­so­ci­ate the sys­tem of poly­no­mi­al equa­tions $$\begin{array}{}\text{(18)}& \prod _{j=1}^N{z}_j^{A_{ij}}=(-1{)}^{(A{B}^t{)}_{ii}\phantom{y_{y_y}}}\prod _{j=1}^N(1-z_j{)}^{B_{ij}}(i=1,\dots ,N).\end{array}$$ This is a straight gen­er­al­iz­a­tion of the ori­gin­al NZ equa­tions in the to­po­lo­gic­al set­ting ex­cept pos­sibly for the sign, but we checked in thou­sands of ex­amples us­ing SnapPy [e24] that this is the right sign, and this could pre­sum­ably be proved us­ing the “par­ity con­di­tion” in [2]. An­oth­er spe­cial case of (18) is the Nahm equa­tion (17), at least when the mat­rix $a$ is in­teg­ral and even, the half-sym­plect­ic mat­rix then be­ing $(1a)$ and the full sym­plect­ic mat­rix $$(\begin{array}{cc}1& a\\ 0& 1\end{array}).$$

For any solu­tion $z$ of (18), the ele­ment $2\sum _{j=1}^N[z_j]$ be­longs to the usu­al Bloch group by the same ar­gu­ment as was used for Nahm sums. We can in fact di­vide by 2 and lift to an ele­ment $[H,z]$ of the ex­ten­ded Bloch group by set­ting $$\begin{array}{}\text{(19)}& [H,z]=\sum _{j=1}^N[u_j,v_j]+[\xi ,{\xi }^{\prime }]+[-\xi ,{\xi }^{\prime }-\xi +\pi i],\end{array}$$ where $u_j$ and $v_j$ are ar­bit­rary choices of log­ar­ithms of $z_j$ and $1-z_j$, re­spect­ively, $\xi$ is defined as $$\frac{1}{\pi }(Au-Bv{)}^t(Cu-Dv)$$ for any com­ple­tion $(CD)$of $H$ to a full sym­plect­ic mat­rix, and ${\xi }^{\prime }$ is any choice of log­ar­ithm of $1-e^{\xi }$. The facts that this is in the ker­nel of $d:\mathbb{Z}[\hat{\mathbb{C}}]\to {\mathsf{Li}}^2(\mathbb{C})$ and that its im­age mod­ulo ex­ten­ded five-term re­la­tions is in­de­pend­ent of all choices (at least mod­ulo 8-tor­sion in the Bloch group of the field gen­er­ated by the $z$’s) can be checked by dir­ect com­pu­ta­tions which are sketched in Sec­tion 6.1 of [e33], to­geth­er with a more pre­cise form that elim­in­ates the tor­sion am­bi­gu­ity.

We have the five fol­low­ing equi­val­ence re­la­tions among pairs, each mo­tiv­ated by a change of the choices made in the to­po­lo­gic­al situ­ation, that do not change this class:

• Sta­bil­ity: in­crease $N$ by $N+1$, re­place $A$ and $B$ in $H=(AB)$ by their dir­ect sums with $(1)$ and $(0)$, re­spect­ively, and set $z_{N+1}=1$, cor­res­pond­ing in the to­po­lo­gic­al case to adding a de­gen­er­ate sim­plex to a 3-man­i­fold tri­an­gu­la­tion.
• Chan­ging the equa­tions: mul­tiply $H$ on the left by an ele­ment of ${\mathrm{GL}}_N(\mathbb{Z})$ without chan­ging the $z$’s. This cor­res­ponds to re­pla­cing the $N$ re­la­tions (18) by mul­ti­plic­at­ive com­bin­a­tions of them in an in­vert­ible way.
• Re­num­ber­ing: mul­tiply $H$ on the right by an $N\times N$ per­muta­tion mat­rix, and per­mute the $z_j$’s by the same mat­rix, cor­res­pond­ing to a re­num­ber­ing of the sim­plices of a tri­an­gu­la­tion.
• New shape para­met­ers: for each $j=1,\dots ,N$, mul­tiply the $N\times 2$ mat­rix $(A_j{B}_j)$ (where $A_j$ and $B_j$ de­note the $j$-th columns of $A$ and $B$, re­spect­ively) by a power of the ele­ment $$(\begin{array}{cc}0& -1\\ 1& -1\end{array})$$ of or­der 3 in ${\mathrm{SL}}_2(\mathbb{Z})$, and re­place $z_j$ cor­res­pond­ingly by $z_j$, $z_j^{\prime }=1/(1-z_j)$, or $z_j^{\prime \prime }=1/(1-z_j)$.
• Al­geb­ra­ic 2-3 Pach­ner moves: re­move two columns of $A$ and the same two columns of $B$ and re­place them by three new columns that are spe­cif­ic $\mathbb{Z}$-lin­ear com­bin­a­tions, sim­ul­tan­eously re­pla­cing $N$ by $N+1$ and chan­ging two of the $z_j$ to three oth­ers in such a way that the cor­res­pond­ing change of $[H,z]$ is a five-term re­la­tion. The ex­pli­cit for­mu­las were first writ­ten down in the spe­cial case cor­res­pond­ing to the Nahm sums (16) by Sander Zwegers in an un­pub­lished 2011 con­fer­ence talk and were then giv­en for ar­bit­rary sym­plect­ic matrices in equa­tion (3-27) of [e20]. This cor­res­ponds to sta­bil­iz­ing $H$ three times and then mul­tiply­ing it on the left by a spe­cif­ic ele­ment of ${\mathrm{Sp}}_{10}(\mathbb{Z})$, and then un­stabil­iz­ing three times.

This gives us a new abeli­an group that maps to the ex­ten­ded Bloch group, namely the set of all pairs $(H,z)$ as above mod­ulo these equi­val­ence re­la­tions, with ad­di­tion giv­en by dir­ect sum. In fact this map is an iso­morph­ism, mean­ing that any ele­ment of the Bloch group can be real­ized by some half-sym­plect­ic mat­rix and solu­tion of the cor­res­pond­ing gen­er­al­ized Neu­mann–Za­gi­er equa­tions and that any five-term re­la­tion can be lif­ted to one com­ing from an al­geb­ra­ic 2-3 Pach­ner move. A more com­plete dis­cus­sion is giv­en in Sec­tion 6.1 of [e33] (full de­tails will be giv­en later), while Sec­tion 6.3 of the same pa­per shows how to at­tach Nahm sum-like $q$-series to ar­bit­rary half-sym­plect­ic matrices.

Nahm’s con­jec­ture already high­lighted a con­nec­tion between half-sym­plect­ic matrices and ques­tions of mod­u­lar­ity, but there are oth­er and more dir­ect con­nec­tions between hy­per­bol­ic 3-man­i­folds and the mod­u­lar group ${\mathrm{SL}}_2(\mathbb{Z})$ that we now de­scribe.

At the end of Sec­tion 5 we dis­cussed Kashaev’s volume con­jec­ture and its re­fine­ment (13). That state­ment in turn was gen­er­al­ized in [e19] on the basis of nu­mer­ic­al com­pu­ta­tions to a con­jec­tur­al asymp­tot­ic for­mula hav­ing a strong mod­u­lar fla­vor. To state it, we first note that the Kashaev in­vari­ant $⟨K{⟩}_n$ of a knot $K$ can be gen­er­al­ized to a func­tion $J^K:\mathbb{Q}/\mathbb{Z}\to \bar{\mathbb{Q}}$ whose value at $-1/n$ for any $n\ge 1$ is $⟨K{⟩}_n$ and which is $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$-equivari­ant. Then the con­jec­tur­al gen­er­al­iz­a­tion of (13) is the state­ment that $$\begin{array}{}\text{(20)}& J^K(\frac{an+b}{cn+d})\sim (cn+d{)}^{3/2}{e}^{{\mathrm{Vol}}_{\mathbb{C}}(K)(n+d/c)/2\pi i}{\mathrm{\Phi }}_{a/c}^K(\frac{2\pi i}{c(cn+d)})J^K(n)\end{array}$$ for every mat­rix $$(\begin{array}{cc}a& b\\ c& d\end{array})\in {\mathrm{SL}}_2(\mathbb{Z})$$ as $n$ tends to in­fin­ity through either in­tegers or ra­tion­al num­bers with bounded de­nom­in­at­or, with (13) be­ing the spe­cial case when $$(\begin{array}{cc}a& b\\ c& d\end{array})=(\begin{array}{cc}0& -1\\ 1& \phantom{-}0\end{array})$$ and $n$ is in­teg­ral. Here ${\mathrm{\Phi }}_{\alpha }^K(h)$ for $\alpha \in \mathbb{Q}/\mathbb{Z}$ is a power series that is con­jec­tured to be the one con­struc­ted in [e31] and dis­cussed at the end of Sec­tion 5.