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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 nonmonogenicfree 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 ℓ 2spaces. 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.
Original language  English 

Pages (fromto)  3753–3793 
Number of pages  41 
Journal  International Mathematics Research Notices 
Volume  2020 
Issue number  12 
Early online date  13 Jun 2018 
DOIs  
Publication status  Published  Jun 2020 
Profiles

Robert Gray
 School of Mathematics  Reader in Pure Mathematics
 Algebra and Combinatorics  Member
 Logic  Member
Person: Research Group Member, Academic, Teaching & Research
Projects
 1 Finished