### Abstract

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