Growth of family of finite simple groups

TL;DR

This paper investigates the growth properties of a specific family of finite simple groups, extending concepts from infinite group growth to finite cases and providing new calculations for linear fractional groups.

Contribution

It introduces the concept of word growth for families of finite groups and computes the growth of finite linear fractional groups, highlighting similarities and differences with infinite groups.

Findings

Computed the word growth of finite linear fractional groups.

Identified parallels and contrasts with infinite group growth.

Extended growth theory to finite simple groups.

Abstract

We consider the growth of an infinite family of finite groups. We are motivated by the remarkable contribution of Bass, Wolf, Milnor, Gromov, Grigorchuk on the word growth and structure of infinite groups, and the results of Black on the word growth of an infinite family of finite groups. We follow the definition of the word growth for families of finite groups as given by Black, and compute the growth of a family of finite linear fractional groups. Some developments are analogous to infinite cases. However, contrasts are also transcribed, as well as other results.