Classifying Spaces from Ore Categories with Garside Families

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 methods we introduce the group Braided T and show that it is of type F∞. Our main object of study are groups that arise as the fundamental group of an Ore category with a Garside family. The two basic motivating examples are the braid groups Braidn and Thompson’s group F . We provide tools to construct classifying spaces with good finiteness properties for these groups. Our first main result can be formulated as follows (see Section 1 for definitions and Section 3 for the general version). Theorem A. Let C be a small right-Ore category that is factor-finite, let ∆ be a right-Garside map, and let ∗ ∈ Ob(C). There is a contractible simplical complex X on which G = π1(C, ∗) acts. The space is covered by the G-translates of compact subcomplexes Kx, x ∈ Ob(C). Every stabilizer is isomorphic to a finite index subgroup of C×(x, x) for some x ∈ C. Taking C to be a Garside monoid and ∆ to be the Garside element, one immediately recovers the known fact that Garside groups, and braid groups in particular, have finite classifying spaces [CMW04]. In fact, if C is taken to be the dual braid monoid, the quotient G\X is precisely Brady’s classifying space for Braidn [Bra01]. In the case of Thompson’s group F the complex in Theorem A is the Stein–Farley complex. The action is not cocompact in this case because C has infinitely many objects. In order to obtain cocompact actions on highly connected spaces, we employ Morse theory. Theorem B. Let C, ∆, ∗ be as in Theorem A and let ρ : Ob(C) → N be a height function such that {x ∈ Ob(C) | ρ(x) ≤ n} is finite for every n ∈ N. Assume that ( stab) C×(x, x) is of type Fn for all x, ( lk) |E(x)| is (n− 1)-connected for all x with ρ(x) beyond a fixed bound. Then π1(C, ∗) is of type Fn. Date: October 10, 2017. 2010 Mathematics Subject Classification. Primary 57M07; Secondary 20F65, 20F36.