by Jason Behrstock
Walter has had a significant impact on my life since I was a graduate student. I’ve continued to learn from him as a mentor and our relationship evolved as we became collaborators and friends. We first met at the weekly “Magnus seminar” which Gilbert Baumslag ran every Friday evening at CUNY and which was typically followed by a dinner of Indian food in one of the nearby restaurants which was a welcoming place for all seminar participants, even the most junior. Gilbert had been an early student of Walter’s father, so they had known each other for most of Walter’s life and seeing the warm relationship they had, as well as the mathematical community they both fostered, was clearly a Neumann family legacy. I thank Walter for this (as well as many other) contributions to our mathematical world and I hope that this short survey of the quasi-isometric classification of fundamental groups of 3-manifolds and, in particular, Walter’s contributions to this area will bring others into the community that he has helped foster and grow throughout his career.
The word metric on a finitely generated group, although nonunique — as it depends on a choice of finite generating set — is canonical when considered up to quasi-isometry (i.e., maps of bounded multiplicative and additive distortion). A fundamental question in group theory, as discussed in Gromov [e2], is to classify finitely generated groups up to quasi-isometry.
An important and rich family of groups are provided by fundamental groups of 3-manifolds. While the geometry of 3-manifolds has been extensively studied following Thurston’s work in the 1970s, the quasi-isometric geometry of these groups has also received significant attention.
One way in which the quasi-isometric geometry of a family of (finitely generated) groups, \( \mathcal G \), can be studied is by breaking it down into two parts:
- quasi-isometric rigidity: if \( G^{\prime} \) is quasi-isometric \( G\in\mathcal G \) must \( G^{\prime}\in\mathcal G \); and,
- quasi-isometric classification: given \( G,G^{\prime}\in\mathcal G \) determine whether \( G \) is quasi-isometric to \( G^{\prime} \).
As we will explain below, for fundamental groups of compact 3-manifolds (henceforth, just called 3-manifold groups), the answer to the first question is “yes,” and the second question is now mostly (but not completely!) known as well.
To begin, we recall that by work of Milnor and Kneser, any 3-manifold admits an essentially canonical decomposition as a connected sum of irreducible 3-manifolds, which are ones in which every 2-dimensional embedded sphere bounds an embedded 3-dimensional ball. Thus, by work of Papasoglu and Whyte on quasi-isometries of free-product decompositions [e10], the study of quasi-isometry types of general 3-manifolds reduces to considering the irreducible ones.
Geometric 3-manifolds are ones which arise as the quotient, by a discrete group of isometries, of one of the eight simply connected homogeneous spaces: \( \mathbb{S}^3 \), \( \mathbb{E}^3 \), \( \mathbb{H}^3 \), \( \mathbb{S}^2\times \mathbb{E} \), \( \mathbb{H}^2\times \mathbb{E} \), \( \mathrm{Nil} \), \( \mathrm{Sol} \), and \( \widetilde{\mathrm{PSL}} \). Perelman’s Geometrization Theorem shows that any irreducible 3-manifold (with zero Euler characteristic) can be cut along tori and Klein bottles (the JSJ decomposition) into pieces which admit geometric structures; when the collection of tori and Klein bottles is nonempty, such a manifold is called nongeometric.
The quasi-isometric classification in the case of geometric closed manifolds is an immediate consequence of the Milnor–Švarc Lemma which provides that the fundamental group of a compact Riemannian manifold is quasi-isometric to the universal of the manifold, and whence in the closed case to one of the eight homogeneous spaces listed above. Note that Rieffel proved that \( \widetilde{\mathrm{PSL}} \) and \( \mathbb{H}^2\times \mathbb{E}^1 \) are quasi-isometric [e9], and thus there are only seven quasi-isometry types of closed geometric 3-manifold groups. For geometric 3-manifolds with boundary the quasi-isometric and commensurability classifications agree: for the hyperbolic case this is a deep theorem of R. Schwartz [e6]; the Seifert fibered space case was first proven by Kapovich and Leeb [e7]. For someone delving into the quasi-isometric classification of 3-manifold groups for the first time, I recommend Section 5 of [e8] as an excellent place to start.
Quasi-isometric rigidity in each of the eight geometric cases were established separately. Note that for geometric 3-manifolds with boundary, only three possible geometries occur: hyperbolic, \( \mathbb{H}^2\times \mathbb{E} \), and \( \widetilde{\mathrm{PSL}} \). For the closed geometric cases see: [e3], [e11], [e4], [e1], [e8], [e5], [e9], [e6].
The first results in the nongeometric case were due to Kapovich and Leeb who settled the quasi-isometric rigidity problem for 3-manifolds with (possibly empty) toroidal boundary (and made the first progress on the classification problem) by showing that quasi-isometries preserve the decomposition into geometric pieces and that quasi-isometries preserve the presence of hyperbolic components [e8], [e7], [e8]. Haïssinsky and Lecuire recently extended the quasi-isometric rigidity result to apply to all 3-manifold groups [e12].
Thus it remains to consider the quasi-isometric classification question. A graph manifold is an irreducible 3-manifold with nontrivial JSJ decomposition in which none of the geometric components admits a hyperbolic structure. Based on their prior work, Kapovich and Leeb asked about the quasi-isometric classification of fundamental groups of closed graph manifolds [e8].
Neumann and I proved the following, of which a special case resolves their question by showing that any two closed nongeometric graph manifolds have bilipschitz homeomorphic universal covers and hence, in particular, quasi-isometric fundamental groups.
- The universal covers, \( \widetilde M \) and \( \widetilde M^{\prime} \), are bilipschitz homeomorphic.
- \( \pi_1(M) \) and \( \pi_1(M^{\prime}) \) are quasi-isometric.
- The Bass–Serre trees for \( M \) and \( M^{\prime} \) are isomorphic as two-colored trees. (Where the vertex groups of \( \pi_{1}(M) \), resp. \( \pi_1(M^{\prime}) \), are colored corresponding to whether the associated Seifert fibered pieces does or does not contain boundary components of \( M \), resp. \( M^{\prime} \).)
- The minimal two-colored graphs in the bisimilarity classes of the decomposition graphs associated to \( M \) and \( M^{\prime} \) are isomorphic. (Again, the vertices are colored corresponding to whether the associated Seifert fibered piece does or does not contain boundary components of \( M \), resp. \( M^{\prime} \).)
Bisimilarity, a notion which arises in computer science, is an algorithmically checkable equivalence relation on colored finite graphs. Each equivalence class has a unique, canonical element which we call minimal. One can list the minimal two-colored graphs of small size, using Theorem 1 this allows us to conclude that, for instance, there are exactly \( 2, 6, 26, 199, 2811, 69711, 2921251, 204535126,\dots \) quasi-isometry classes of fundamental groups of nongeometric graph manifolds composed of at most \( 1,2,3,4,5,6,7,8,\dots \) Seifert fibered pieces [1] (see also oeis.org/A120388).
For nongeometric manifolds with hyperbolic pieces, Neumann and I discovered a similar, albeit more intricate, classification in terms of bisimilarity of certain labeled graphs. One issue which arises in obtaining the quasi-isometric classification is choosing a canonical model for each (universal cover of a) geometric component, by choosing the sizes of the horoballs in the model which is chosen uniformly for the entire commensurability class. In the case where the pieces are nonarithmetic, this can be done by making a choice in the commensurability quotient; in the arithmetic case, by work of Margulis, the commensurator subgroup is not discrete, so making a choice of model is more complicated (and one that we weren’t able to treat in all cases). For nongeometric manifolds with only hyperbolic pieces of which at least one is nonarithmetic (we call these NAH-manifolds), the classifying objects are also minimal labeled graphs, but now the edges are labeled as well as the vertices, and the labelings are more complicated: each vertex is labeled by the isomorphism type of an orientable, complete, hyperbolic orbifold and each edge is labeled by a linear isomorphism between certain 2-dimensional \( \mathbb Q \)-vector spaces. We call such a graph an NAH-graph. Bisimilarity for NAH-graphs is defined similarly as for two-colored graphs, namely these are open graph-homomorphisms which preserve labels in a controlled way. In the case of such manifolds, Neumann and I proved:
In certain cases we related the commensurability and quasi-isometric classification of NAH-manifolds. One version of this is given in the following, which makes use of Neumann’s strengthening of Leighton’s theorem [2].
The original version of the above theorem was conditional on the 3–dimensional version of the following conjecture about the structure of hyperbolic manifolds. The 3–dimensional version was proven by Wise in ([e13], Corollary 16.15), but the general version remains an interesting open question:
