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."