Lower bound on growth of non-elementary subgroups in relatively hyperbolic groups

TL;DR

This paper establishes a linear lower bound on the logarithmic growth rate of non-elementary subgroups in relatively hyperbolic groups, demonstrating their uniform exponential growth.

Contribution

It provides the first explicit lower bound on subgroup growth in relatively hyperbolic groups, linking subgroup size to growth rate.

Findings

Non-elementary subgroups exhibit uniform exponential growth.

Logarithm growth rate is linearly bounded by the generating set size.

Results apply to all non-elementary subgroups in the class.

Abstract

This paper proves that in a non-elementary relatively hyperbolic group, the logarithm growth rate of any non-elementary subgroup has a linear lower bound by the logarithm of the size of the corresponding generating set. As a consequence, any non-elementary subgroup has uniform exponential growth.