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Classifying spaces from Ore categories with Garside families. (English) Zbl 1444.57016
The paper in review formalizes a “blueprint” to prove finiteness properties of Thompson groups. There are many results known, which usually follow along the same lines as the proofs of K. S. Brown [J. Pure Appl. Algebra 44, 45–75 (1987; Zbl 0613.20033)], M. Stein [Trans. Am. Math. Soc. 332, No. 2, 477–514 (1992; Zbl 0798.20025)] and D. S. Farley [Topology 42, No. 5, 1065–1082 (2003; Zbl 1044.20023)].
The article in review provides a theorem that reduces a statement about finiteness properties of Thompson groups to its technical core, which is about connectivity of certain complexes.
The proof is formulated categorically using the Ore category and Garside family.
The following theorems are stated and results are proven in greater generality later on:
(Theorem A) Let \(\mathcal{C}\) be a small right-Ore category that is factor-finite and admits a right-Garside map \(\Delta\), and let \(* \in \text{Ob}(\mathcal{C})\). There is a contractible simplicial complex \(X\) on which \(G = \pi_1(\mathcal{C}, *)\) acts. The space is covered by the \(G\)-translates of compact subcomplexes \(K_x\) for \(x \in \text{Ob}(\mathcal{C})\). Every stabilizer is isomorphic to a finite-index subgroup of the automorphism group \(\mathcal{C}^\times(x,x)\) for some \(x \in \text{Ob}(\mathcal{C})\).
(Theorem B) Let \(\mathcal{C}, \Delta, *\) be as in (Theorem A) and let \(\rho: \text{Ob}(\mathcal{C}) \to \mathbb{N}\) be a height function such that \(\{x \in \text{Ob}(\mathcal{C}\,|\, \rho(x) \leq n\}\) is finite for every \(n \in \mathbb{N}\). Assume that
(STAB) \(\mathcal{C}^\times(x,x)\) is of type \(F_n\) for all \(x\),
(LK) there exists an \(N \in \mathbb{N}\) such that \(|E(x)|\) is \((n-1)\)-connected for all \(x\) with \(\rho(x) \geq N\).
Then \(\pi_1(\mathcal{C}, *)\) is of type \(F_n\).
The braided Thompson group \(BT\), braided \(T\), is defined and the following theorem is shown: The braided Thompson group BT is of type \(F_\infty\).

MSC:
57M07 Topological methods in group theory
20F36 Braid groups; Artin groups
20F65 Geometric group theory
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