Finiteness-property conjecture for commutator subgroups of thick buildings

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Let d∈Nd \in \mathbb{N} and let (\Delta_n)_{n \in \mathbb{N}) be a sequence of finite dd-dimensional buildings whose thickness satisfies

th⁡(Δn)→∞\operatorname{th}(\Delta_n) \rightarrow \infty

as n→∞n \rightarrow \infty. For each building Δn\Delta_n, let WΔnW_{\Delta_n} be its right-angled Coxeter group and let WΔn′W_{\Delta_n}' be the commutator subgroup. The finiteness-property conjecture. Then, for all but finitely many nn, there is a kernel of a map

WΔn′→ZW_{\Delta_n}'\to\mathbb{Z}

that is of type F⁡d\operatorname{F}_d but not of type FP⁡d+1\operatorname{FP}_{d+1}. This would give groups arising from sufficiently thick buildings with a sharp gap between finiteness properties F⁡d\operatorname{F}_d and FP⁡d+1\operatorname{FP}_{d+1}. The statement is presented as expected from the preceding spherical-subcomplex conjecture and is not resolved in the source.

References

Primary source

Eduard Schesler and Matthew C. B. Zaremsky, “Random subcomplexes of finite buildings, and fibering of commutator subgroups of right-angled Coxeter groups”, arXiv:2107.10958 (2022).

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