Affiliation:
1. School of Mathematics, University of East Anglia, Norwich NR4 7TJ, UK
2. School of Mathematics, University of Manchester, Manchester M13 9PL, UK
Abstract
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 nonmonogenic-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 Følner condition suffices for the spectral radius to be at least $1$; since left amenability implies the weak Følner 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.
Funder
Engineering and Physical Sciences Research Council
Publisher
Oxford University Press (OUP)
Cited by
2 articles.
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