Random semigroups á la Gromov

Abstract

We introduce a model of random semigroups analogous to Gromov's density model for random groups. For a fixed number $n\ge 2$ of generators and a density parameter $δ\in (0,1)$, we impose a uniformly chosen set of $\lfloor n^{2δl}\rfloor$ relations of the form $u=v$, where $u,v$ are distinct words of length at most $l$, and study the resulting semigroup as $l\to\infty$. We prove that, with overwhelming probability, above density $1/2$ the semigroup is trivial, while below density $1/2$ it contains a free subsemigroup of rank two, has a Golod-Shafarevich semigroup algebra over every field, and has a homomorphic image which generates a non-elementary hyperbolic group. We also show that the small cancellation condition $C(m)$ holds with overwhelming probability whenever $δ<1/2m$. Finally, for $n\ge 3$, at critical density $1/2$ we obtain a subsequential limiting distribution on finite semigroups, with positive mass at the trivial semigroup and unbounded support in size. This contrasts with random groups, which have order at most two with overwhelming probability at critical density.