Daniel Scott Farley

Finiteness and CAT(0) properties of diagram groups

Abstract Any diagram group over a finite semigroup presentation acts properly, freely, and cellularly by isometrices on a proper CAT(0) cubical complex. The existence of a proper, cellular action by isometries on a CAT(0) cubical complex has powerful consequences for the acting group G. One gets, for example, a proof that G satisfies the Baum–Connes conjecture. Any diagram group over a finite presentation of a finite semigroup is of type F ∞ .

Discrete Morse theory and graph braid groups

If is any finite graph, then the unlabelled configuration space of n points on , denoted U C n , is the space of n-element subsets of . The braid group of on n strands is the fundamental group of U C n . We apply a discrete version of Morse theory to these U C n , for any n and any , and provide a clear description of the critical cells i n every case. As a result, we can calculate a presentation for the braid group of any tree, for any number of strands. We also give a simple proof of a theorem due to Ghrist: the space U C n strong deformation retracts onto a CW complex of dimension at most k, where k is the number of vertices in of degree at least 3 (and k is thus independent of n). AMS Classification 20F65, 20F36; 57M15, 57Q05, 55R80

Proper Isometric Actions of Thompson's Groups on Hilbert Space

In 1965, Thompson defined the groups F, T , and V [8]. Thompson’s group V is the group of right-continuous bijections v of [0, 1] that map dyadic rational numbers to dyadic rational numbers, that are differentiable except at finitely many dyadic rational numbers, and such that, on each interval on which v is differentiable, v is affine with derivative a power of 2. The group F is the subgroup of V consisting of homeomorphisms. The group T is the subgroup of V consisting of those elements which induce homeomorphisms of the circle, where the circle is regarded as [0, 1] with 0 and 1 identified. It is a long-standing open question to determine whether F is amenable. The main theorem of this paper establishes that the groups F, T , and V all have the weaker property of a-T-menability. A theorem of Higson and Kasparov [4] states that every a-T-menable group satisfies the Baum-Connes conjecture with arbitrary coefficients, so Thompson’s groups F, T , and V satisfy the Baum-Connes conjecture as well. An isometric action of a discrete group G on a metric space X is proper if, for any x ∈ X and any bounded subset U of X, there are only finitely many elements of g that translate x inside U. A function f : V1 → V2 between two vector spaces is affine if it is the composition of a linear map followed by a translation.

On the cohomology rings of tree braid groups

Abstract Let Γ be a finite connected graph. The (unlabelled) configuration space U C n Γ of n points on Γ is the space of n -element subsets of Γ . The n -strand braid group of Γ , denoted B n Γ , is the fundamental group of U C n Γ . We use the methods and results of [Daniel Farley, Lucas Sabalka, Discrete Morse theory and graph braid groups, Algebr. Geom. Topol. 5 (2005) 1075–1109. Electronic] to get a partial description of the cohomology rings H ∗ ( B n T ) , where T is a tree. Our results are then used to prove that B n T is a right-angled Artin group if and only if T is linear or n 4 . This gives a large number of counterexamples to Ghrist’s conjecture that braid groups of planar graphs are right-angled Artin groups.

Homological and finiteness properties of picture groups

Picture groups are a class of groups introduced by Guba and Sapir. Known examples include Thompsons groups F, T, and V. In this paper, a large class of picture groups is proved to be of type F∞. A Morse-theoretic argument shows that, for a given picture group, the rational homology vanishes in almost all dimensions.

Constructions of E and Eℱℬ for groups acting on CAT(0) spaces

If G is a group acting properly by semisimple isometries on a proper CAT(0) space X, then we build models for the classifying spaces E_{vc} and E_{fbc} under the additional assumption that the action of G has a well-behaved collection of axes in X. (This hypothesis is described in the paper.) We conjecture that the latter hypothesis is satisfied in a large range of cases. Our classifying spaces resemble those created by Connolly, Fehrman, and Hartglass for crystallographic groups G.

THE ACTION OF THOMPSON'S GROUP ON A CAT(0) BOUNDARY

For a given locally finite CAT(0) cubical complex X with base vertex �, we define the profile of a given geodesic ray c issuing fromto be the collection of all hyperplanes (in the sense of (17)) crossed by c. We give necessary conditions for a collection of hyperplanes to form the profile of a geodesic ray, and conjecture that these conditions are also sufficient. We show that profiles in diagram and picture complexes can be expressed naturally as infinite pictures (or diagrams), and use this fact to describe the fixed points at infinity of the actions by Thompsons groups F, T, and V on their respective CAT(0) cubical complexes. In particular, the actions of T and V have no global fixed points. We obtain a partial description of the fixed set of F; it consists, at least, of an arc c of Tits length �/2, and any other fixed points of F must have one particular profile, which we describe. We conjecture that all of the fixed points of F lie on the arc c. Our results are motivated by the problem of determining whether F is amenable.

A proof that Thompson's groups have infinitely many relative ends

Abstract Each of Thompsons groups F, T, and V has infinitely many ends relative to the groups F [0, 1/2], T [0, 1/2], and V [0, 1/2) (respectively). We can therefore simplify the proof, due to Napier and Ramachandran, that F, T, and V are not Kähler groups. Thompsons groups T and V have Serres property FA. The original proof of this fact was due to Ken Brown.

Quasiautomorphism groups of type

The groups QF, QT, and QV are groups of quasi-automorphisms of the infinite binary tree. Their names indicate a similarity with Thompsons well-known groups F, T, and V. We will use the theory of diagram groups over semigroup presentations to prove that all of the above groups (and several generalizations) have type F-infinity. Our proof uses certain types of hybrid diagrams, which have properties in common with both planar diagrams and braided diagrams. The diagram groups defined by hybrid diagrams also act properly and isometrically on CAT(0) cubical complexes.

F_\infty

We prove that any hyperplane H in a CAT(0) cubical complex X has no self-intersections and separates X into two convex complementary components. These facts were originally proved by Sageev. Our argument shows that his theorem is a corollary of Gromov’s link condition. We also give new arguments establishing some combinatorial properties of hyperplanes. We show that these properties are sufficient to prove that the 0-skeleton of any CAT(0) cubical complex is a discrete median algebra, a fact that was previously proved by Chepoi, Gerasimov, and Roller.