Abstract
We study the connection between amenability, Følner conditions and the geometry of finitely generated semigroups. Using results of Klawe, we show that within an extremely broad class of semigroups (encompassing all groups, left cancellative semigroups, finite semigroups, compact topological semigroups, inverse semigroups, regular semigroups, commutative semigroups and semigroups with a left, right or twosided zero element), left amenability coincides with the strong Følner condition. Within the same class, we show that a finitely generated semigroup of subexponential growth is left amenable if and only if it is left reversible. We show that the (weak) Følner condition is a left quasiisometry invariant of finitely generated semigroups, and hence that left amenability is a left quasiisometry invariant of left cancellative semigroups. We also give a new characterisation of the strong Følner condition in terms of the existence of weak Følner sets satisfying a local injectivity condition on the relevant translation action of the semigroup.
Original language  English 

Pages (fromto)  80878103 
Number of pages  17 
Journal  Transactions of the American Mathematical Society 
Volume  369 
DOIs  
Publication status  Published  1 May 2017 
Profiles

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