Quasi-isometry invariance of hyperbolicity in semimetric spaces, digraphs and semigroups

TL;DR

This paper proves that a specific geometric property called hyperbolicity remains invariant under quasi-isometries in certain semimetric spaces, addressing an open problem in semigroup theory.

Contribution

It establishes the quasi-isometry invariance of hyperbolicity in semimetric spaces like digraphs and semigroups under a mild geometric condition, partially solving an open problem.

Findings

Hyperbolicity is preserved under quasi-isometries with an additional geometric assumption.

Addresses and partially solves a problem posed by Gray and Kambites.

Extends the understanding of geometric invariants in semigroup and digraph contexts.

Abstract

Gray and Kambites introduced a notion of hyperbolicity in the setting of semimetric spaces like digraphs or semigroups. We will prove that under a small additional geometric assumption their notion of hyperbolicity is preserved by quasi-isometries. Applied to semigroups, this will partially solve a problem of Gray and Kambites.