 
Summary: Journal of Topology Page 1 of 7 c 2007 London Mathematical Society
doi:10.1112/jtopol/jtm010
Heegaard genus and property for hyperbolic 3manifolds
D. D. Long, A. Lubotzky and A. W. Reid
Abstract
We show that any finitely generated nonelementary Kleinian group has a cofinal family of finite index normal
subgroups with respect to which it has Property . As a consequence, any closed hyperbolic 3manifold has a
cofinal family of finite index normal subgroups for which the infimal Heegaard gradient is positive.
1. Introduction
Let M be a finite volume hyperbolic 3manifold and L = {Mi} some family of finite sheeted
regular covers of M. We say that L is cofinal if i 1(Mi) = {1}, where, as usual, the 1(Mi)
are all to be regarded as subgroups of 1(M). The infimal Heegaard gradient of M with respect
to the family L is defined as:
infi
h
(Mi)
[1(M) : 1(Mi)]
,
where h
(Mi) denotes the minimal value for the negative of the Euler characteristic of a
