Uniform growth rate

TL;DR

This paper establishes a universal upper bound on the growth rate of outcomes in certain evolutionary systems, including groups like the mapping class group and triangulation classes, based on local mutation rules.

Contribution

It introduces a general principle for bounding growth rates in systems with local mutation rules and applies it to groups and surface triangulations.

Findings

Derived a uniform exponential upper bound for group growth rates.

Established a similar bound for triangulation homotopy classes.

Demonstrated the bound's independence from initial object size.

Abstract

In an evolutionary system in which the rules of mutation are local in nature, the number of possible outcomes after $m$ mutations is an exponential function of $m$ but with a rate that depends only on the set of rules and not the size of the original object. We apply this principle to find a uniform upper bound for the growth rate of certain groups including the mapping class group. We also find a uniform upper bound for the growth rate of the number of homotopy classes of triangulations of an oriented surface that can be obtained from a given triangulation using $m$ diagonal flips.