Geometry, Topology and Dynamics Seminar
Jing Tao
University of Oklahoma
Growth tight actions and strongly contracting elements
Abstract: Let $G$ be a finitely-generated non-elementary group acting on a based metric space $(X,d,o)$. Let
$B(o,r)$ be the number of $G$-orbits in a ball of radius $r$ about $o$. The growth rate of $G$
relative to $X$ is $h(G,X) = \limsup_{r \to \infty} \log B(o,r) / r$. For any normal subgroup $N$ of $G$,
one can similarly define the growth rate of $G/N$ relative to $X/N$. In general, $h(G/N,X/N)$ is no
bigger than $h(G,N)$. An action $G$ on $X$ is called growth tight if for all infinite normal subgroups $N$,
$h(G/N, X/N) < h(G,N)$. In this talk, I will explain a result that shows that if $G$ contains a
strongly-contracting element and if $G$ is not too "badly distorted" in $X$, then the action of $G$ on $X$ is
growth tight. This generalizes all previous known results about growth tightness of cocompact actions.
This is joint work with Goulnara Arzhantseva and Chris Cashen.
Monday March 17, 2014 at 3:00 PM in SEO 636