Date:
Thu, 22/01/202610:00-11:00
Location:
Ross 70
Abstract:
Background. Hanna Neumann (1957) conjectured that for subgroups H and J of a free group, rk(H∩J) - 1 ≤ (rk(H) - 1)(rk(J) - 1). This famous conjecture (HNC) was proved only in 2011. Wise (2003) gave an analogous conjecture for rank-1 subgroups, i.e. for words; it was proved by Helfer–Wise and independently by Louder–Wilton, who recast it as a gap phenomenon for a word invariant: the stable primitivity rank. Ernst-West, Puder and Seidel (“EWPS”, 2023) introduced a q-analogue of this invariant and conjectured the same gap phenomenon. In seemingly unrelated work, Reiter (2019) studied fixed points of words under random homomorphisms into finite permutation groups, and conjectured a tight bound.
New results. We develop a theory of polymatroids on graphs. A polymatroid is a concept from probability and information theory, generalizing entropy. We prove a gap theorem for such polymatroids. As special cases, we prove Reiter’s conjecture, prove the EWPS gap conjecture, and obtain new proofs of Wise’s conjecture and its analogs. We also show that some closely related invariants of words and subgroups in free groups are profinitely rigid, via fixed-point counts in stable actions of finite simple groups of large rank. Finally, we conjecture a q-analogue of the HNC and suggest that many other sequences of finite group actions may have a corresponding version of the HNC. arXiv:2601.00053.
Background. Hanna Neumann (1957) conjectured that for subgroups H and J of a free group, rk(H∩J) - 1 ≤ (rk(H) - 1)(rk(J) - 1). This famous conjecture (HNC) was proved only in 2011. Wise (2003) gave an analogous conjecture for rank-1 subgroups, i.e. for words; it was proved by Helfer–Wise and independently by Louder–Wilton, who recast it as a gap phenomenon for a word invariant: the stable primitivity rank. Ernst-West, Puder and Seidel (“EWPS”, 2023) introduced a q-analogue of this invariant and conjectured the same gap phenomenon. In seemingly unrelated work, Reiter (2019) studied fixed points of words under random homomorphisms into finite permutation groups, and conjectured a tight bound.
New results. We develop a theory of polymatroids on graphs. A polymatroid is a concept from probability and information theory, generalizing entropy. We prove a gap theorem for such polymatroids. As special cases, we prove Reiter’s conjecture, prove the EWPS gap conjecture, and obtain new proofs of Wise’s conjecture and its analogs. We also show that some closely related invariants of words and subgroups in free groups are profinitely rigid, via fixed-point counts in stable actions of finite simple groups of large rank. Finally, we conjecture a q-analogue of the HNC and suggest that many other sequences of finite group actions may have a corresponding version of the HNC. arXiv:2601.00053.