A quasi-isometric realization of virtual isomorphism

We construct a continuous reduction from virtual isomorphism to quasi-isometry on the space of all finitely generated marked groups. Virtual isomorphism allows passage to finite-index subgroups and quotients by finite normal subgroups. The reduction has a fixed finite number of output generators and includes finite and trivial inputs. It establishes the forward direction of the strict comparison asked by Simon Thomas in 2003. We first represent virtual isomorphism by conjugate commensurability of subgroups of a fixed free group. Abelian attachments then realize this subgroup relation by quasi-isometry. The new attachments absorb arbitrary finite-index changes and recover the variable subgroup even when its conjugates have proper coarse inclusions. GitHub repository: isomorphism-quasi-isometry