Relative Subgroup Growth and Subgroup Distortion

TL;DR

This paper investigates the relationship between subgroup growth and distortion in finitely generated groups, providing conditions for extending length functions and characterizing possible growth and distortion functions.

Contribution

It establishes connections between relative subgroup growth and distortion functions, and characterizes which functions can arise as these invariants.

Findings

Identifies conditions for extending length functions between groups

Characterizes functions equivalent to subgroup growth and distortion

Explores the relationship between subgroup growth and distortion functions

Abstract

We study the relative growth of finitely generated subgroups in finitely generated groups, and the corresponding distortion function of the embeddings. We explore which functions are equivalent to the relative growth functions and distortion functions of finitely generated subgroups. We also study the connections between these two asymptotic invariants of group embeddings. We give conditions under which a length function on a finitely generated group can be extended to a length function on a larger group.