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We prove that an infinite-ended group whose one-ended factors havefinite-index subgroups and are in a family of groups with a nonzeromultiplicative invariant is not quasi-isometrically rigid. Combining thisresult with work of the first author proves that a residually-finitemulti-ended hyperbolic group is quasi-isometrically rigid if and only if it isvirtually free. The proof adapts an argument of Whyte for commensurability offree products of closed hyperbolic surface groups.

Failure of quasi-isometric rigidity for infinite-ended groups