Virtual-surjection conjectures for discrete groups

For a subdirect product of groups of type Fₖ, do finite-index projections onto every k-tuple force the subgroup to be of type Fₖ? The homological analogue replaces Fₖ by FPₖ.

Why it matters and what counts as progress

Why it matters: Virtual surjection connects finiteness of many coordinate projections with finiteness of the whole subgroup.

What counts as progress: Proofs of both discrete-group conjectures, or a counterexample; related homological fibre-product results must retain the common quotient’s finite-presentability hypothesis.

How the problem developed

13 Jul 2011 · Conjecture [1]
Benno Kuckuck Formulated the virtual-surjection conjecture: for a subdirect product of groups of type F_n, finite-index projections onto every n-tuple should force the subgroup to be type F_n.
20 Jul 2026 · Reported claim [2]
Tal Cohen and Mark Shusterman Reported proofs of the discrete-group virtual-surjection conjectures for both F_k and FP_k; their homological n-(n+1)-(n+2) result assumes the common quotient is FP and gives counterexamples when that assumption is dropped.

AI contribution

Virtual Surjection and the n-(n+1)-(n+2) Theorem for Discrete Groups

Tal Cohen, Mark Shusterman

Cohen and Shusterman prove the Fₖ and FPₖ virtual-surjection conjectures for discrete groups. Their homological fibre-product theorem assumes the common quotient Q is finitely presented; they also give counterexamples without that condition.

AI involvementAI-centered

ImportanceNotable

Technical verificationFormal (human)

Human understandingFull