Celebratio Mathematica

In the pa­per “Ex­po­nen­tial de­cay of cor­rel­a­tion coef­fi­cients for geodes­ic flows”1 Moore proved the ex­po­nen­tial mix­ing of the geodes­ic flow on hy­per­bol­ic sur­faces (later this res­ult has been gen­er­al­ized to ana­log­ous flows on sym­met­ric spaces). Be­low we briefly ex­plain how we use this res­ult in the proofs of the sur­face sub­group the­or­em2 and the Ehren­pre­is con­jec­ture3  — the lat­ter now a the­or­em.

The sur­face sub­group the­or­em states that there is a ${\pi }_1$-in­ject­ive im­mersed sur­face in every closed hy­per­bol­ic 3-man­i­fold $M$. We built such a sur­face out of nearly iso­met­ric­ally im­mersed “$R$-pants”, which are hy­per­bol­ic pairs of pants with geodes­ic bound­ar­ies of length $R$. We re­quire $R$ to be large and we find these pants in $M$ by first tak­ing two tri­pods, where a tri­pod is three vec­tors in a tan­gent plane at a point in $M$, spaced at an angle of $2\pi /3$ from each oth­er. We then look for three nearly geodes­ic con­nec­tions between the two tri­pods; for each triple of con­nec­tions, there is a pair of pants in $M$ of the de­sired form.

We want to know that these pants are evenly dis­trib­uted around each closed geodes­ic $\gamma$ in $M$ of suit­able length (close to $R$). (We can then match them with each oth­er to form a closed and nearly geodes­ic sur­face.) Giv­en two tri­pods with two con­nec­tions between them, where the two con­nec­tions to­geth­er are ho­mo­top­ic to $\gamma$, we can freely move the tri­pods and con­nec­tions around $\gamma$, and hence these doubly con­nec­ted tri­pods are evenly dis­trib­uted. It fol­lows from the ex­po­nen­tial mix­ing of the frame flow that the num­ber of third con­nec­tions (between the re­main­ing vec­tor of each tri­pod) is roughly the same no mat­ter where the tri­pods are; there­fore the triply con­nec­ted tri­pods are evenly dis­trib­uted around $\gamma$.

It’s worth say­ing a bit more about ex­actly how we use ex­po­nen­tial mix­ing. We ac­tu­ally take weighted sums of triply con­nec­ted tri­pods, where the weight is the product of the weights for each of the three con­nec­tions, which is com­puted as fol­lows. Around each tan­gent vec­tor in the tri­pod, which we ca­non­ic­ally equip with a frame, we take a stand­ard bump func­tion in the frame bundle $\mathcal{F}M$ of $M$, and then let the weight for the con­nec­tion be $${\int }_{\mathcal{F}\tilde{M}}(g_R{)}_{\ast }({\tilde{\phi }}_{-}){\tilde{\phi }}_{+},$$ where the ${\tilde{\phi }}_{\pm }$ are the bump func­tions, lif­ted by the con­nec­tion to the uni­ver­sal cov­er $\tilde{M}$ of $M$, and $g_R$ is the frame flow. The point is then that the total weight of all con­nec­tions between the two vec­tors is $${\int }_{\mathcal{F}M}(g_R{)}_{\ast }({\phi }_{-}){\phi }_{+},$$ and this is guar­an­teed to be (ex­po­nen­tially) close to a con­stant by ex­po­nen­tial mix­ing.

To prove what was formerly known as the Ehren­pre­is con­jec­ture, we must cor­rect the dis­crep­ancy between the total weight of pants on the two sides of a closed geodes­ic on a giv­en hy­per­bol­ic sur­face. By the reas­on­ing above, this dis­crep­ancy is ex­po­nen­tially small, and we show that any dis­crep­ancy can be cor­rec­ted by a modi­fic­a­tion of weights that is only a poly­no­mi­al (in $R$) mul­tiple of the dis­crep­ancy. Hence the needed modi­fic­a­tion is small, and can be made without mak­ing any weight neg­at­ive.

The fact that the geodes­ic/frame flows are mix­ing for neg­at­ively curved man­i­folds is well known but in­suf­fi­cient for the above ap­plic­a­tions. What is cru­cially used above is that the flow has ex­po­nen­tial de­cay of cor­rel­a­tions, the fact Moore proved us­ing Gel­fand’s the­ory of rep­res­ent­a­tion of (cer­tain) Lie groups.

Vladi­mir Marković is a Pro­fess­or of Math­em­at­ics at the Uni­versity of Ox­ford, and a Fel­low of the Roy­al So­ci­ety. Jeremy Kahn is a Pro­fess­or of Math­em­at­ics at Brown Uni­versity. To­geth­er they were awar­ded the 2012 Clay Re­search Award for their proofs of the Sur­face Sub­group The­or­em and the Ehren­pre­is Con­jec­ture, and also gave an In­vited Ad­dress at the 2014 In­ter­na­tion­al Con­gress of Math­em­aticians.