On Belk's classifying space for Thompson's group F
Lucas Sabalka, Matthew C. B. Zaremsky
备注：
v2 is a significant rewrite of v1. An error was found in Lemma 4.5 of v1, invalidating the last 25% of the paper. As such, v2 comprises the first part of v1, with some related necessary revisions. The main result about Thompson's group F remains unchanged. v2 has 12 pages and 4 figures
The space of configurations of n ordered points in the plane serves as a classifying space for the pure braid group PB_n. Elements of Thompson's group F admit a model similar to braids, except instead of braiding the strands split and merge. In Belk's thesis, a space CF was considered, of configurations of points on the real line allowing for splitting and merging, and a proof was sketched that CF is a classifying space for F. The idea there was to build the universal cover and construct an explicit contraction to a point. Here we start with an established CAT(0) cube complex X on which F acts freely, and construct an explicit homotopy equivalence between X/F and CF, proving that CF is indeed a K(F,1).
In this paper we generalize techniques of Belk-Matucci to solve the conjugacy problem for every Thompson-like group $V_n(H)$, where $n \geq 2$ and $H$ is a subgroup of the symmetric group on $n$ elements. We use this to prove that, if $n \neq m$, $V_n(H)$ is not isomorphic to $V_m(G)$ for any $H,G$.
We prove that Richard Thompson's group F is not minimally almost convex with respect to the two standard generators. This improves upon a recent result of S. Cleary and J. Taback. We make use of the forest diagrams for elements of F introduced by J. Belk and K. Brown. These diagrams seem particularly well-suited for understanding the Cayley graph for the two standard generators.
We show that the Basilica Thompson group introduced by Belk and Forrest is not finitely presented, and in fact is not of type FP_2. The proof involves developing techniques for proving non-simple connectedness of certain subcomplexes of CAT(0) cube complexes.
This note serves as a short and reader-friendly introduction to twisted Brin-Thompson groups, which were recently constructed by Belk and the author to provide a family of simple groups with a variety of interesting properties. Most notably, twisted Brin-Thompson groups can be used to show that every finitely generated group quasi-isometrically embeds as a subgroup of a finitely generated simple group. Another important application is a concrete construction of a family of simple groups with arbitrary finiteness length. In addition to giving a concise introduction to the groups and these appli…
We prove that Claas Röver's Thompson-Grigorchuk simple group $V\mathcal{G}$ has type $F_\infty$. The proof involves constructing two complexes on which $V\mathcal{G}$ acts: a simplicial complex analogous to the Stein complex for $V$, and a polysimiplical complex analogous to the Farley complex for $V$. We then analyze the descending links of the polysimplicial complex, using a theorem of Belk and Forrest to prove increasing connectivity.
We define a family of groups that generalises Thompson's groups $T$ and $G$ and also those of Higman, Stein and Brin. For groups in this family we descrine centralisers of finite subgroups and show, that for a given finite subgroup $Q$, there are finitely many conjugacy classes of finite subgroups isomorphic to $Q$. We use this to show that whenever defined, the T versions of these groups have a slightly weaker property, quasi-$\underline{\operatorname F}_\infty$, to that of a group possessing a finite type model for the classifying space for proper actions ${\underline{E}}G$ if and only if th…
We study centralisers of finite order automorphisms of generalisations of Thompson's group F and conjugacy classes of finite subgroups in finite extensions of these groups. In particular, we show that centralisers of finite automorphisms in these generalised Thompson groups either admit a finite type classifying space or are not finitely generated. As an application we deduce results about the Bredon type of such finite extensions.
By the density of a finite graph we mean its average vertex degree. For an $m$-generated group, the density of its Cayley graph in a given set of generators, is the supremum of densities taken over all its finite subgraphs. It is known that a group with $m$ generators is amenable iff the density of the corresponding Cayley graph equals $2m$. A famous problem on the amenability of R.\,Thompson's group $F$ is still open. What is known due to the result by Belk and Brown, is that the density of its Cayley graph in the standard set of group generators $\{x_0,x_1\}$, is at least $3.5$. This estimat…
We introduce forest diagrams and strand diagrams for elements of Thompson's group F. A forest diagram is a pair of infinite, bounded binary forests together with an order-preserving bijection of the leaves. Using forest diagrams, we derive a simple length formula for elements of F, and we discuss applications to the geometry of the Cayley graph, including a new upper bound on the isoperimetric constant (a.k.a. Cheeger constant) of F. Strand diagrams are similar to tree diagrams, but they can be concatenated like braids. Motivated by the fact that configuration spaces are classifying spaces for…
We describe how an Ore category with a Garside family can be used to construct a classifying space for its fundamental group(s). The construction simultaneously generalizes Brady's classifying space for braid groups and the Stein--Farley complexes used for various relatives of Thompson's groups. It recovers the fact that Garside groups have finite classifying spaces. We describe the categories and Garside structures underlying certain Thompson groups. The Zappa--Szép product of categories is introduced and used to construct new categories and groups from known ones. As an illustration of our m…
We build actions of Thompson group V (related to the Cantor set) and of the so-called "spheromorphism" group of Neretin, on "towers" of moduli spaces of genus zero real stable curves. The latter consist of inductive limits of spaces which are the real parts of the Grothendieck-Knudsen compactification of the usual moduli spaces of punctured Riemann spheres. By a result of M. Davis, T. Januszkiewicz and R. Scott, these spaces are aspherical cubical complexes, whose fundamental groups, the "pure quasi-braid groups", are some analogues of the classical pure braid groups. By lifting the actions of…
We show that diagram groups can be viewed as fundamental groups of spaces of positive paths on directed 2-complexes (these spaces of paths turn out to be classifying spaces). Thus diagram groups are analogs of second homotopy groups, although diagram groups are as a rule non-Abelian. Part of the paper is a review of the previous results from this point of view. In particular, we show that the so called rigidity of the R.Thompson's group $F$ and some other groups is similar to the flat torus theorem. We find several finitely presented diagram groups (even of type F_\infty) each of which contain…
