The large scale geometry of inverse semigroups and their maximal group images

The geometry of inverse semigroups is a natural topic of study, motivated both from within semigroup theory and by applications to the theory of non-commutative $C^*$-algebras. We study the relationship between the geometry of an inverse semigroup and that of its maximal group image, and in particular the geometric \textit{distortion} of the natural map from the former to the latter. This turns out to have both implications for semigroup theory and potential relevance for operator algebras associated to inverse semigroups. Along the way, we also answer a question of Lled\'{o} and Mart\'{i}nez by providing a more direct proof that an $E$-unitary inverse semigroup has Yu's Property A if its maximal group image does.