Thang Le (Georgia Institute of Technology)
Title: "Growth of torsion homology in finite coverings"
Abstract: "Suppose X is a finite CW complex and (X_n) is a sequence of
finite coverings of X. We will discuss the growth of the torsion part of
the homology groups of X_n with integer coefficients. When the
coverings are abelian, under some assumption on the exhaustion of the
covering, we show that the growth of the torsion homology is given by
the Mahler measure of the first non-zero (multivariable) Alexander
polynomial of the corresponding homology group of the univerval abelian
covering. When X is a hyperbolic 3-manifold, we show the growth of the
torsion homology is bounded from above by the hyperbolic volume, and it
is conjectured that this bound is exact. It turns out that this
conjecture, if proved, is stronger than a recent result of Kojima and
McShane on the relation between the dilitation of a pseudo-Anosov
homeomorphisms and the hyperbolic volume of the mapping torus, and also
a conjecture of McMullen, now proved by Liu and Hadari, about the
homological dilitation of pseudo-Anosov homeomorphisms."