On growth, identities and free subsemigroups for inverse semigroups of deficiency one

TLDR: It is proved that for any positive integer n there exists an inverse semigroup ϒn of deficiency 1 and rank n + 1 such that ϒ n has exponential growth and it does not contain nonmonogenic free inverse subsemigroups.

Abstract: For any positive integer n > 1 we construct an example of inverse semigroup with n generators and n - 1 defining relations which has cubic growth and at least n generators in any presentation. This semigroup has the same set of identities as the free monogenic inverse semigroup. In particular, we give the first example of a one relation nonmonogenic inverse semigroup having polynomial growth. We also prove that for any positive integer n there exists an inverse semigroup ϒn of deficiency 1 and rank n + 1 such that ϒn has exponential growth and it does not contain nonmonogenic free inverse subsemigroups. Furthermore, ϒn satisfies the identity [[x, y], [z, t]]2 = [[x, y], [z, t]] of quasi-solvability and it contains a free subsemigroup of rank 2.