# On cogrowth, amenability and the spectral radius of a random walk on a semigroup

## TL;DR

This paper introduces two notions of cogrowth for finitely generated semigroups, explores their relationship with amenability and random walks, and establishes new conditions under which semigroups are amenable.

## Contribution

It defines local and global cogrowth for semigroups, characterizes free semigroups via minimal global cogrowth, and links cogrowth with spectral radius and amenability.

## Key findings

- Maximal global or local cogrowth implies spectral radius ≥ 1.
- Maximal local cogrowth implies amenability in left reversible semigroups.
- Maximal global cogrowth suffices for left amenability in broad classes of semigroups.

## Abstract

We introduce two natural notions of cogrowth for finitely generated semigroups --- one local and one global --- and study their relationship with amenability and random walks. We establish the minimal and maximal possible values for cogrowth rates, and show that non-monogenic free semigroups are exactly characterised by minimal global cogrowth. We consider the relationship with cogrowth for groups and with amenability of semigroups. We also study the relationship with random walks on finitely generated semigroups, and in particular the spectral radius of the associated Markov operators (when defined) on $\ell_2$-spaces. We show that either of maximal global cogrowth or the weak F{\o}lner condition suffices for its spectral radius to be at least $1$, since left amenability implies the weak F{\o}lner condition, this represents a generalisation to semigroups of one implication of Kesten's Theorem for groups. By combining with known results about amenability, we are able to establish a number of new sufficient conditions for (left or right) amenability in broad classes of semigroups. In particular, maximal local cogrowth left implies amenability in any left reversible semigroup, while maximal global cogrowth (which is a much weaker property) suffices for left amenability in an extremely broad class of semigroups encompassing all inverse semigroups, left reversible left cancellative semigroups and left reversible regular semigroups.