by Jeremy Kahn and Vladimir Marković
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 \[ \int_{\mathcal{F} \tilde M} (g_R)_*(\tilde\phi_-) \tilde\phi_+, \] 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 \[ \int_{\mathcal{F} M} (g_R)_*(\phi_-) \phi_+, \] 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.
