On a curious problem and what it lead to · 2022-lie-156

Source-reported assessment from the ICM catalogue snapshot. This is not an independent review by Frontier Science.

The problem is unresolved as of 3 August 2026. No proof or counterexample is known for all infinite finitely presented residually finite groups with property $(T)$ and all residual normal chains. The decisive gap is that the rank gradient is the cost minus one of the particular free profinite action associated with the chain, whereas the known theorem that Kazhdan groups have cost one concerns the infimum of costs and does not force every such profinite action to have cost one.

Key references and progress:

• Miklós Abért, Andrei Jaikin-Zapirain, and Nikolay Nikolov, “The rank gradient from a combinatorial viewpoint,” Groups, Geometry, and Dynamics 5 (2011), 213–230, DOI 10.4171/GGD/124. Conjecture 10 asks for vanishing rank gradient for infinite property-$(T)$ groups along every normal chain with trivial intersection. The paper proves vanishing results for other classes, including amenable groups and groups with an infinite soluble normal subgroup, but not the Kazhdan case.

• Miklós Abért and Nikolay Nikolov, “Rank gradient, cost of groups and the rank versus Heegaard genus problem,” Journal of the European Mathematical Society 14 (2012), 1657–1677, DOI 10.4171/JEMS/344. Their cost correspondence gives

for the boundary action of a Farber chain. A normal chain with trivial intersection is Farber, so the exact problem asks whether all of these profinite actions have cost one.

• Mark Shusterman, “Ranks of subgroups in boundedly generated groups,” Bulletin of the London Mathematical Society 48 (2016), 539–547, DOI 10.1112/blms/bdw023. Theorem 1.2 gives sublinear upper bounds for the ranks of arbitrary finite-index subgroups of a boundedly generated group (and a logarithmic-squared bound for normal subgroups). Consequently the assertion holds for the boundedly generated groups within the class in the problem, but bounded generation is an additional hypothesis.

• Tom Hutchcroft and Gábor Pete, “Kazhdan groups have cost 1,” Inventiones Mathematicae 221 (2020), 873–891, DOI 10.1007/s00222-020-00967-6. Theorem 1.2 proves that every countably infinite Kazhdan group has cost one, equivalently that it admits a free ergodic p.m.p. action of cost one. The authors explicitly note that fixed price one remains open and that their construction is not a factor-of-i.i.d. construction; therefore it does not determine the costs of the profinite actions attached to the chains in this problem.

• Miklós Abért, “On a curious problem and what it lead to,” Proceedings of the International Congress of Mathematicians 2022, vol. 5 (2022), 3374–3387, Problem 11, DOI 10.4171/ICM2022/181. This states the exact finitely presented, residually finite property-$(T)$ problem and records it as open after the Hutchcroft–Pete theorem.

• Mikołaj Frączyk, Sam Mellick, and Amanda Wilkens, “Poisson–Voronoi tessellations and fixed price in higher rank,” arXiv preprint (2023), arXiv:2307.01194. They prove fixed price one for lattices in higher-rank semisimple real Lie groups and in products of at least two automorphism groups of regular trees. Combined with the Abért–Nikolov correspondence, this proves the required vanishing for all residual normal chains in the property-$(T)$ lattices covered by their theorem.

• Raz Slutsky, “A Product-Neighbourhood Criterion for Fixed Price One,” arXiv preprint (2026), arXiv:2607.20273. This extends fixed price one to higher-rank lattices over local fields, automorphism groups and lattices of affine buildings (including exotic buildings), and lattices in products of two noncompact compactly generated unimodular locally compact groups. It yields further property-$(T)$ special cases, but its geometric/product hypotheses do not exhaust the groups in the problem.

Thus the strongest progress proves the assertion for broad arithmetic, higher-rank, building, product, and boundedly generated subclasses. What remains open is the universal case for property-$(T)$ groups outside these additional structural hypotheses; in particular, the literature does not show that every free profinite action arising from an arbitrary residual normal chain has cost one.