The Lubotzky-Sarnak Conjecture asserts that the fundamental group of a
finite volume hyperbolic manifold does not have Property \tau.
Put in a geometric context, this conjecture predicts a tower of finite sheeted covers for which the Cheeger constant goes to zero. This conjecture has attracted a lot of attention recently because of its connections to the topology of finite sheeted covers of closed hyperbolic 3-manifolds. This talk will discuss these connections, together with recent work that connects this circle of ideas with the group theoretic property LERF (a far reaching generalization of residually finite).
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