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algebra / Geometric group theory

The Virtual Surjection Conjecture for Discrete Groups

If a subgroup of a product of groups of type $F_k$ virtually surjects onto every $k$-tuple of factors, must it be of type $F_k$ itself? Yes, for discrete groups, and likewise for $FP_k$. The homological $n$-$(n+1)$-$(n+2)$ Conjecture follows for discrete groups when the common quotient is finitely presented, and that hypothesis cannot be dropped.

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algebraJul 20, 2026Significance 20/100Registry: unreviewed

The Virtual Surjection Conjecture for Discrete Groups

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If a subgroup of a product of groups of type $F_k$ virtually surjects onto every $k$-tuple of factors, must it be of type $F_k$ itself? Yes, for discrete groups, and likewise for $FP_k$. The homological $n$-$(n+1)$-$(n+2)$ Conjecture follows for discrete groups when the common quotient is finitely presented, and that hypothesis cannot be dropped.

Source authenticatedThe Virtual Surjection Conjecture for Discrete Groups →
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If a subgroup of a product of groups of type $F_k$ virtually surjects onto every $k$-tuple of factors, must it be of type $F_k$ itself? Yes, for discrete groups, and likewise for $FP_k$. The homological $n$-$(n+1)$-$(n+2)$ Conjecture follows for discrete groups when the common quotient is finitely presented, and that hypothesis cannot be dropped.

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The Virtual Surjection Conjecture for Discrete Groups — Mathematical Frontier Network