Free coarse groups

Abstract

A coarse group is a group endowed with a coarse structure so that the group multiplication and inversion are coarse mappings. Let $(X,\mathcal{E})$ be a coarse space and let $\mathfrak{M}$ be a variety of groups different from the variety of singletons. We prove that there is a coarse group $F_{\mathfrak{M}}(X,\mathcal{E})\in \mathfrak{M}$ such that $(X,\mathcal{E})$ is a subspace of $F_{\mathfrak{M}}(X,\mathcal{E})$, $X$ generates $F_{\mathfrak{M}}(X,\mathcal{E})$ and every coarse mapping $(X,\mathcal{E})⟶(G,{\mathcal{E}}^{\prime })$ where $G\in \mathfrak{M}$, $(G,{\mathcal{E}}^{\prime })$ is a coarse group, can be extended to coarse homomorphism $F_{\mathfrak{M}}(X,\mathcal{E})⟶(G,{\mathcal{E}}^{\prime })$. If $\mathfrak{M}$ is the variety of all groups, the groups $F_{\mathfrak{M}}(X,\mathcal{E})$ are asymptotic counterparts of Markov free topological groups over Tikhonov spaces.

Coarse space, coarse group, variety, free coarse group, Markov free topological group