On the Complexity of the Cogrowth Sequence

TL;DR

This paper investigates the complexity of the cogrowth sequence in finitely generated groups, proving it is not P-recursive for certain amenable groups with superpolynomial growth, thus addressing a key open question.

Contribution

It establishes that the cogrowth sequence is not P-recursive for amenable groups of superpolynomial growth, providing new insights into the sequence's computational complexity.

Findings

Cogrowth sequence is not P-recursive for certain groups.

Addresses an open question by Garrabant and Pak.

Links group growth properties to sequence complexity.

Abstract

Given a finitely generated group with generating set $S$, we study the \emph{cogrowth} sequence, which is the number of words of length $n$ over the alphabet $S$ that are equal to one. This is related to the probability of return for walks in a Cayley graph with steps from $S$. We prove that the cogrowth sequence is not $P$-recursive when~$G$ is an amenable group of superpolynomial growth, answering a question of Garrabant and Pak.