Growth tightness for groups with contracting elements

TL;DR

This paper proves that groups with contracting elements, including non-elementary relatively hyperbolic groups and CAT(0) groups with rank-1 elements, exhibit growth tightness, extending previous results in geometric group theory.

Contribution

It establishes growth tightness for a broad class of groups with contracting elements, generalizing prior work and connecting boundary properties to growth behavior.

Findings

Non-elementary relatively hyperbolic groups are growth tight.

Groups with nontrivial Floyd boundary are growth tight.

CAT(0) groups with rank-1 elements are growth tight.

Abstract

We establish growth tightness for a class of groups acting geometrically on a geodesic metric space and containing a contracting element. As a consequence, any group with nontrivial Floyd boundary are proven to be growth tight with respect to word metrics. In particular, all non-elementary relatively hyperbolic group are growth tight. This generalizes previous works of Arzhantseva-Lysenok and Sambusetti. Another interesting consequence is that CAT(0) groups with rank-1 elements are growth tight with respect to CAT(0)-metric.