Among cocompact special groups, being linearly polynomially hyperbolic is equivalent to not containing F2 × F2 as a subgroup, rendering the latter a quasi-isometric invariant.
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Polynomial hyperbolicity and products of free groups
Among cocompact special groups, being linearly polynomially hyperbolic is equivalent to not containing F2 × F2 as a subgroup, rendering the latter a quasi-isometric invariant.