Avramidi–Delzant higher-rank bounded-cohomological-dimension conjecture

Establish whether, for every integer n, there exists a constant f(n) such that any group G generated by at most n elements acting isometrically on a symmetric space X of curvature -1 <= kappa <= 0 and real rank r, with displacement greater than f(n), satisfies cd_K G <= r for every field K; in particular, determine the validity of the Avramidi–Delzant group-theoretic conjecture beyond the counterexample for X = H^2 x H^2 exhibited in the paper.

Background

Avramidi and Delzant proved that a finitely generated group acting on a hyperbolic space with sufficiently large minimal displacement must be free. They proposed a higher-rank analogue for groups acting on symmetric spaces: sufficiently large displacement, combined with a bounded number of generators, should force the cohomological dimension over every field to be no greater than the real rank of the symmetric space.

The paper presents finite-index subgroups of the fundamental group of S x S, where S is a closed genus-two hyperbolic surface, that have uniformly bounded rank, arbitrarily large displacement on H2 x H2, and cohomological dimension four. Since H2 x H2 has real rank two, these examples disprove the conjecture in that specific setting. The conjecture is nevertheless the explicitly stated unresolved/general research claim recorded in the introduction, subject to this counterexample.

References

They conjecture a higher-rank analogue Conjecture~4. We state the following group-theoretic consequence of their conjecture. Indeed, as explained in , applying Conjecture 4 there to the augmentation ideal of K[G] gives the stated bound on $\operatorname{cd}_K G$.

\begin{conjecture}[Avramidi--Delzant, group-theoretic form]\label{conj-AD} Let $X$ be a symmetric space with curvature $-1 \le \kappa \le 0$ of real rank $r$. For every $n$ there exists a constant $f(n)$ with the following property: Let $G$ be a group with $d(G) \le n$. Suppose $G$ acts on $X$ isometrically such that \operatorname{disp}_X(G)>f(n). Then, $\operatorname{cd}_K G\le r $ for every field $K$. \end{conjecture}

— Finite covers of a product of surfaces with bounded rank and arbitrarily large systole  (2609.21952 - Fujiwara, 18 Sep 2026) in Introduction, the displayed conjecture labeled “Avramidi--Delzant, group-theoretic form”