arXiv subjects

Daniel Farley

Publications and source records attributed to Daniel Farley.

14 recordsLinked to original sources

arXiv2025

Simple Expansion Sets and Non-Positive Curvature

An expansion set is a set $\mathcal{B}$ such that each $b \in \mathcal{B}$ is equipped with a set of expansions $\mathcal{E}(b)$. The theory of expansion sets offers a systematic approach to the construction of classifying spaces for generalized Thompson groups. We say that $\mathcal{B}$ is simple if proper expansions are unique when they exist. We will prove that any given simple expansion set determines a cubical complex with a metric of non-positive curvature. In many cases, the cubical complex will be CAT(0). We are thus able to recover proofs that Thompsons groups $F$, $T$, and $V$, Houghton's groups $H_{n}$, and groups defined by finite similarity structures all act on CAT(0) cubical complexes. We further state a sufficient condition for the cubical complex to be locally finite, and show that the latter condition is satisfied in the cases of $F$, $T$, $V$, and $H_{n}$.

math.GR↗

arXiv2023

Finiteness properties of generalized Thompson groups via expansion sets

We outline a general procedure that builds classifying spaces for generalized Thompson groups $\Gamma$. The construction depends on a small number of choices: (1) an inverse semigroup $S$ of partial transformations that ``locally determine" $\Gamma$; (2) an equivalence relation on certain pairs $(f,D)$, and (3) an ``expansion" rule $\mathcal{E}$. These choices determine an \emph{expansion set} $\mathcal{B}$, which is a combinatorial device that outputs a simplicial complex $\Delta^{f}_{\mathcal{B}}$ upon which $\Gamma$ acts. Under favorable conditions, often achieved in practice, $\Delta^{f}_{\mathcal{B}}$ is contractible, and the action of $\Gamma$ has small stabilizers. The definition of $\Delta^{f}_{\mathcal{B}}$ is such that ascending and descending links in $\Delta^{f}_{\mathcal{B}}$ can be described via formulas that depend only on the expansion rule $\mathcal{E}$. The result is to facilitate the usual computations of the connectivity of the descending link. Under natural hypotheses, one can prove that the acting group has type $F_{\infty}$. The net effect of our results is to automate results of this kind. Several applications are given; in particular, we sketch unified proofs that $V$, $nV$, R\"{o}ver's group $G$, and the Lodha-Moore group have type $F_{\infty}$.

math.GR↗

arXiv2022

Finiteness properties of some groups of piecewise projective homeomorphisms

The Lodha-Moore group $G$ first arose as a finitely presented counterexample to von Neumann's conjecture. The group $G$ acts on the unit interval via piecewise projective homemorphisms. A result of Lodha shows that $G$ in fact has type $F_{\infty}$. Here we will describe $G$ as a group that is "locally determined" by an inverse semigroup $S_{2}$, in the sense of the author's joint work with Hughes. The semigroup $S_{2}$ is generated by three linear fractional transformations $A$, $B$, and $C_{2}$, where $A$ and $B$ are elliptical transformations of the hyperbolic plane and $C_{2}$ is a hyperbolic translation. Following a general procedure delineated by Farley and Hughes, we offer a new proof that $G$ has type $F_{\infty}$. Our proof simultaneously shows that various groups acting on the line, the circle, and the Cantor set have type $F_{\infty}$. We also prove analogous results for the groups that are locally determined by an inverse semigroup $S_{3}$, which shares the generators $A$ and $B$ with $S_{2}$, but replaces $C_{2}$ with a different hyperbolic translation $C_{3}$.

math.GR↗

arXiv2014

Braided Diagram Groups and Local Similarity Groups

Hughes defined a class of groups that act as local similarities on compact ultrametric spaces. Guba and Sapir had previously defined braided diagram groups over semigroup presentations. The two classes of groups share some common characteristics: both act properly by isometries on CAT(0) cubical complexes, and certain groups in both classes have type F-infinity, for instance. Here we clarify the relationship between these families of groups: the braided diagram groups over tree-like semigroup presentations are precisely the groups that act on compact ultrametric spaces via small similarity structures. The proof can be considered a generalization of the proof that Thompson's group V is a braided diagram group over a tree-like semigroup presentation. We also prove that certain additional groups, such as the Houghton groups, and a certain group of quasi-automorphisms lie in both classes.

math.GR↗

arXiv2014

Local similarity groups with context-free co-word problem

Let $G$ be a group, and let $S$ be a finite subset of $G$ that generates $G$ as a monoid. The co-word problem is the collection of words in the free monoid $S^{\ast}$ that represent non-trivial elements of $G$. A current conjecture, based originally on a conjecture of Lehnert and modified into its current form by Bleak, Matucci, and Neuhöffer, says that Thompson's group $V$ is a universal group with context-free co-word problem. In other words, it is conjectured that a group has a context-free co-word problem exactly if it is a finitely generated subgroup of $V$. Hughes introduced the class $\mathcal{FSS}$ of groups that are determined by finite similarity structures. An $\mathcal{FSS}$ group acts by local similarities on a compact ultrametric space. Thompson's group $V$ is a representative example, but there are many others. We show that $\mathcal{FSS}$ groups have context-free co-word problem under a minimal additional hypothesis. As a result, we can specify a subfamily of $\mathcal{FSS}$ groups that are potential counterexamples to the conjecture.

math.GR↗

arXiv2012

The Lower Algebraic K-Theory of Split Three-Dimensional Crystallographic Groups

We explicitly compute the lower algebraic K-theory of the split three-dimensional crystallographic groups; i.e., the groups G that act properly and cocompactly on three-dimensional Euclidean space by isometries, such that the natural map from G to O(3) is a split injection onto its image. There are 73 split three-dimensional crystallographic groups in all, out of a total of 219 isomorphism types of three-dimensional crystallographic groups. We also provide a general splitting formula for the lower algebraic K-theory that is valid for all three-dimensional crystallographic groups. This result generalizes earlier work of Alves and Ontaneda. Along the way, we give explicit descriptions of all 73 split three-dimensional crystallographic groups, and completely work out their classification. The split crystallographic groups are sometimes called "splitting groups". A theorem of crystallographic groups says that any crystallographic group is a finite-index subgroup of its splitting group, so each three-dimensional crystallographic group is a finite-index subgroup of one from our list.

math.KT↗

arXiv2009

A proof of Sageev's Theorem on hyperplanes in CAT(0) cubical complexes

We prove that a hyperplane 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 has previously been proved by Chepoi, Gerasimov, and Roller.

math.GT↗

arXiv2009

Presentations of Graph Braid Groups

Let G be a graph. The (unlabeled) configuration space of n points on G is the space of all n-element subsets of G. The fundamental group of such a configuration space is called a graph braid group. We use a version of discrete Morse theory to compute presentations of all graph braid groups, for all finite connected graphs G and all natural numbers n.

math.GR↗

arXiv2009

Constructions of E_{vc} and E_{fbc} 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.

math.AT↗

arXiv2007

A Proof that Thompson's Groups have Infinitely Many Relative Ends

We show that each of Thompson's groups F, T, and V have infinitely many ends relative to certain subgroups. We go on to show that T and V both have Serre's property FA, i.e., any action of T or V on a tree will have a fixed point. (The proof of the latter statement was originally due to Ken Brown, and our proof is based on his notes.)

math.GR↗

arXiv2007

On the cohomology rings of tree braid groups

Let $Γ$ be a finite connected graph. The (unlabelled) configuration space $UC^n Γ$ of $n$ points on $Γ$ is the space of $n$-element subsets of $Γ$. The $n$-strand braid group of $Γ$, denoted $B_nΓ$, is the fundamental group of $UC^n Γ$. We use the methods and results of our paper "Discrete Morse theory and graph braid groups" 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.

math.GR↗

arXiv2006

Presentations for the cohomology rings of tree braid groups

If G is a finite graph and n is a natural number, then the n-strand braid group of G is the fundamental group of the configuration space of n points on G. This article gives a complete computation of the integral cohomology rings of the n-strand braid groups in case G is a tree. Some of the argument builds on earlier joint work with Lucas Sabalka.

math.AT↗

arXiv2006

The Action of Thompson's Group on a CAT(0) Boundary

One way to show that Thompson's group F is non-amenable is to exhibit an action of F on a locally compact CAT(0) space X containing no F-invariant flats and having no global fixed points in its boundary-at-infinity. We study the actions of Thompson's groups F, T, and V on the boundaries-at-infinity of proper CAT(0) cubical complexes. In particular, we show that Thompson's groups T and V act without fixing any points in the boundaries of their CAT(0) cubical complexes. This in particular gives another proof of the well-known fact that these groups are non-amenable. We obtain a partial description of the fixed set for F: Thompson's group F fixes an arc in the boundary of its cubical complex. We leave open the possibility that there are more fixed points, but describe a region of the boundary which must contain all of the others.

math.GR↗

arXiv2005

Discrete Morse theory and graph braid groups

If Gamma is any finite graph, then the unlabelled configuration space of n points on Gamma, denoted UC^n(Gamma), is the space of n-element subsets of Gamma. The braid group of Gamma on n strands is the fundamental group of UC^n(Gamma). We apply a discrete version of Morse theory to these UC^n(Gamma), for any n and any Gamma, and provide a clear description of the critical cells in 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 UC^n(Gamma) strong deformation retracts onto a CW complex of dimension at most k, where k is the number of vertices in Gamma of degree at least 3 (and k is thus independent of n).

math.GR↗