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.
We prove that the Poisson boundary of a simple random walk on the Schreier graph of action of $F$ on $\mathbb{D}$, where $\mathbb{D}$ is the set of dyadic numbers in $[0, 1]$, is non-trivial. This gives a new proof of the result of Kaimanovich: Thompson's group $F$ doesn't have Liouville property. In addition, we compute growth function of the Schreier graph of the action of $F$ on $\mathbb{D}$.
Orderable Thompson-like groups arising from Ore categories
Université de Genève（日内瓦大学）；Université Paris-Saclay（巴黎萨克雷大学）；CNRS（法国国家科学研究中心）；Laboratoire de mathématiques d’Orsay（奥赛数学实验室）；Université de Rennes（雷恩大学）；IRMAR（雷恩数学与应用研究所）
We reformulate the statement of the Feit-Thompson theorem in terms of diagrams in the category of finite groups, namely iterations of the Quillen lifting property with respect to particular morphisms.
In this article, we state and prove a general criterion allowing us to show that some groups are hyperbolically elementary, meaning that every isometric action of one of these groups on a Gromov-hyperbolic space either fixes a point at infinity or has bounded orbits. Also, we show how such a hyperbolic rigidity leads to fixed-point properties on finite-dimensional CAT(0) cube complexes. As an application, we prove that Thompson's group $V$ is hyperbolically elementary, and we deduce that it satisfies Property $(FW_{\infty})$, ie., every isometric action of $V$ on a finite-dimensional CAT(0) cu…
We investigate Farley's CAT(0) cubical model for Thompson's group F (we adopt the classical language of F, using binary trees and piecewise linear maps). Main results include: in general, Thompson's group elements are parabolic; we find simple, exact formulas for the CAT(0) translation lengths, in particular the elements of F are ballistic and uniformly bounded away from zero; there exist flats of any dimension and we construct explicitly many of them; we reveal large regions in the Tits Boundary, for example the positive part of a non-separable Hilbert sphere, but also more complicated object…
Structure monoids and groups are algebraic invariants of equational varieties. We show how to construct presentations of these objects from coherent categorifications of equational varieties, generalising several results of Dehornoy. We subsequently realise the higher Thompson groups $F_{n,1}$ and the Higman-Thompson groups $G_{n,1}$ as structure groups. We go on to obtain presentations of these groups via coherent categorifications of the varieties of higher-order associativity and of higher-order associativity and commutativity, respectively. These categorifications generalise Mac Lane's pen…
The subgroup structure of Thompson's group F is not yet fully understood. The group F is a subgroup of the group PL(I) of orientation preserving, piecewise linear self homeomorphisms of the unit interval and this larger group thus also has a poorly understood subgroup structure. It is reasonable to guess that F is the "only" subgroup of PL(I) that is not elementary amenable. In this paper, we explore the complexity of the elementary amenable subgroups of F in an attempt to understand the boundary between the elementary amenable subgroups and the non-elementary amenable. We construct an example…
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.
We introduce the Pythagorean C*-algebras and use the category/functor method to construct unitary representations of Thompson's groups from representations of them. We calculate several examples.
We show that Thompson's group F is the symmetry group of the "generic idempotent". That is, take the monoidal category freely generated by an object A and an isomorphism A \otimes A --> A; then F is the group of automorphisms of A.
Bieri, Geoghegan and Kochloukova computed the BNSR-invariants $Σ^m(F)$ of Thompson's group $F$ for all $m$. We recompute these using entirely geometric techniques, making use of the Stein--Farley CAT(0) cube complex on which $F$ acts.
We propose a new unified framework for Thompson-like groups using a well-known device called operads and category theory as language. We discuss examples of operad groups which have appeared in the literature before. As a first application, we give sufficient conditions for plain operads to yield groups of type $F_\infty$. This unifies and extends existing proofs for certain Thompson-like groups in a conceptual manner.
We compute the higher $Σ$-invariants $Σ^m(F_{n,\infty})$ of the generalized Thompson groups $F_{n,\infty}$, for all $m,n\ge 2$. This extends the $n=2$ case done by Bieri, Geoghegan and Kochloukova, and the $m=2$ case done by Kochloukova. Our approach differs from those used in the $n=2$ and $m=2$ cases; we look at the action of $F_{n,\infty}$ on a $\textrm{CAT}(0)$ cube complex, and use Morse theory to compute all the $Σ^m(F_{n,\infty})$. We also obtain lower bounds on $Σ^m(H_n)$, for the Houghton groups $H_n$, again using actions on $\textrm{CAT}(0)$ cube complexes, and discuss evidence that …
We give a criterion for group elements to have fixed points with respect to a semi-simple action on a complete CAT(0) space of finite topological dimension. As an application, we show that Thompson's group T and various generalizations of Thompson's group V have global fixed points when they act semi-simply on finite-dimensional complete CAT(0) spaces.
We prove that random walks on Thompson's group $F$ driven by strictly non-degenerate finitely supported probability measures $μ$ have a non-trivial Poisson boundary. The proof consists in an explicit construction of two different non-trivial $μ$-boundaries. Both of them are defined in terms of the Schreier graph $Γ$ on the dyadic-rational orbit of the canonical action of $F$ on the unit interval (actually, we consider a natural embedding of $F$ into the group $PLF({\mathbb R})$ of piecewise linear homeomorphisms of the real line, and realize $Γ$ on the dyadic-rational orbit in ${\mathbb R}$). …
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…
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…
For a finite digraph $D$, we define the corresponding subshift of finite type $(X_D, σ_D)$ to be the dynamical system where $X_D$ is the set of all bi-infinite walks through $D$ and $σ_D$ is the shift operator. Two digraphs $D_1$ and $D_2$ are called shift equivalent if there is $m\geq 1$ such that $(X_{D_1}, σ_{D_1}^n)$ and $(X_{D_2}, σ_{D_2}^n)$ are topologically conjugate for all $n\geq m$. They are called strong shift equivalent if this holds for $m=1$. In this paper we introduce a new category which generalises the category of subshifts of finite type and topological conjugacy. Our catego…
In previous work, joint with Bux, Fluch, Marschler and Witzel, we proved that the braided Thompson groups are of type $\textrm{F}_\infty$. The proof utilized certain contractible cube complexes, which in this paper we prove are CAT(0). We then use this fact to compute the geometric invariants $Σ^m(F_{\textrm{br}})$ of the pure braided Thompson group $F_{\textrm{br}}$. Only the first invariant $Σ^1(F_{\textrm{br}})$ was previously known. A consequence of our computation is that as soon as a subgroup of $F_{\textrm{br}}$ containing the commutator subgroup $[F_{\textrm{br}},F_{\textrm{br}}]$ is f…
We prove that Thompson's group $V$ is acyclic, answering a 1992 question of Brown in the positive. More generally, we identify the homology of the Higman-Thompson groups $V_{n,r}$ with the homology of the zeroth component of the infinite loop space of the mod $n-1$ Moore spectrum. As $V = V_{2,1}$, we can deduce that this group is acyclic. Our proof involves establishing homological stability with respect to $r$, as well as a computation of the algebraic K-theory of the category of finitely generated free Cantor algebras of any type $n$.
The famous 3x + 1 problem of L. Collatz needs no introduction; however, this paper concerns a lesser-known, but similarly unresolved, precursor problem : the Original Collatz Conjecture, or OCC. We demonstrate that the core arithmetic operator from the OCC, when combined with a conjunction of J.-Y. Girard from his Geometry of Interaction system, leads to a realisation of R. Thompson's group F as congruential functions, in the sense of J. Conway. We also give the underlying category theory that accounts for this, and describe the core operator from the OCC as a canonical coherence isomorphism.
In the quest in constructing conformal field theories (CFT) Jones has discovered a beautiful and deep connection between CFT, Richard Thompson's groups and knot theory. This led to a powerful functorial framework for constructing actions of particular groups arising from categories such as Thompson's groups and braid groups. In particular, given a group and two of its endomorphisms one can construct a semidirect product where the largest Thompson's group $V$ is acting. These semidirect products have remarkable diagrammatic descriptions which were previously used to provide new examples of grou…
Maximal ideals of reduced group C*-algebras and Thompson's groups
Lund University（隆德大学）；University of Southern Denmark（南丹麦大学）；Newcastle University（纽卡斯尔大学）；University of Erlangen–Nuremberg（埃尔朗根-纽伦堡大学）；Federal University of Pelotas（佩洛塔斯联邦大学）
We propose a new unifying framework for Thompson-like groups using a well-known device called operads and category theory as language. We discuss examples of operad groups which have appeared in the literature before. As a first application, we proof a theorem which implies that planar or symmetric or braided operads with transformations satisfying some finiteness conditions yield operad groups of type $F_\infty$. This unifies and extends existing proofs that certain Thompson-like groups are of type $F_\infty$.
We give sufficient conditions for the non-triviality of the Poisson boundary of random walks on $H(\mathbb{Z})$ and its subgroups. The group $H(\mathbb{Z})$ is the group of piecewise projective homeomorphisms over the integers defined by Monod. For a finitely generated subgroup $H$ of $H(\mathbb{Z})$, we prove that either $H$ is solvable, or every measure on $H$ with finite first moment that generates it as a semigroup has non-trivial Poisson boundary. In particular, we prove the non-triviality of the Poisson boundary of measures on Thompson's group $F$ that generate it as a semigroup and have…
Vaughan Jones discovered unexpected connections between Richard Thompson's group and subfactor theory while attempting to construct conformal field theories (in short CFT). Among other this founded Jones' technology: a powerful new method for constructing actions of fraction groups which had numerous applications in mathematical physics, operator algebras, group theory and more surprisingly in knot theory and noncommutative probability theory. We propose and outline a program in the vein of Jones' work but where the Thompson group is replaced by a family of groups that we name forest-skein gro…
We consider diagram groups as defined by V. Guba and M. Sapir. A diagram group G acts on the associated cube complex K by isometries. It is known that if a cube complex L is of a finite dimension then every isometry g of L is semi-simple, i.e. its translation length is realized. It was conjectured by D. S. Farley that in the case of a diagram group G the action of G on the associated cube complex K is by semisimple isometries even when K has an infinite dimension. In this paper we give a counterexample to Farley Conjecture and we show that R. Thompson's group F, considered as a diagram group, …
We show that if G is an admissible group acting geometrically on a CAT(0) space X, then G is a hierarchically hyperbolic space and with mild assumptions the sublinearly-Morse boundary of the group is a topological model for associated Poisson boundaries .
We correct Theorem 3.10 of "The simplicial boundary of a CAT(0) cube complex" in the infinite-dimensional case. No correction is needed in the finite-dimensional case.
It is well known that the Tits boundary of a proper cocompact CAT(0) space embeds into every asymptotic cone of the space. We explore the relationships between the asymptotic cones of a CAT(0) space and its boundary under both the standard visual (i.e. cone) topology and the Tits metric. We show that the set of asymptotic cones of a proper cocompact CAT(0) space admits canonical connecting maps under which the direct limit is isometric to the Euclidean cone on the Tits boundary. The resulting projection from any asymptotic cone to the Tits boundary is determined by the visual topology; on the …
If $(W,S)$ is a right-angled Coxeter system and $W$ has no $\mathbb Z^3$ subgroups, then it is shown that the absence of an elementary separation property in the presentation diagram for $(W,S)$ implies all CAT(0) spaces acted on geometrically by $W$ have locally connected CAT(0) boundary. It was previously known that if the presentation diagram of a general right-angled Coxeter system satisfied the separation property then all CAT(0) spaces acted on geometrically by $W$ have non-locally connected boundary. In particular, this gives a complete classification of the right-angled Coxeter groups …
We specify exactly which groups can act geometrically on CAT(0) spaces whose visual boundary is homeomorphic to either a circle or a suspension of a Cantor set.
We investigate the Tits boundary of locally compact CAT(0) 2-complexes. In particular we show that away from the endpoints, a geodesic segment in the Tits boundary is the ideal boundary of an isometrically embedded Euclidean sector. As applications, we provide sufficient conditions for two points in the Tits boundary to be the endpoints of a geodesic in the 2-complex and for a group generated by two hyperbolic isometries to contain a free group. We also show that if two CAT(0) 2-complexes are quasi-isometric then the cores of their Tits boundaries are bi-Lipschitz.
We show that, given any finite dimensional, connected, compact metric space Z, there exists a group G acting geometrically on two CAT(0) spaces X and Y, a G-equivariant quasi-isometry f from X to Y, and a geodesic ray c in X, such that the closure of f(c), instersected with the boundary of Y, is homeomorphic to Z. This characterizes all homeomorphism types of "geodesic boundary images" that arise in this manner.
The Morse boundary of a proper geodesic metric space is designed to encode hypberbolic-like behavior in the space. A key property of this boundary is that a quasi-isometry between two such spaces induces a homeomorphism on their Morse boundaries. In this paper we investigate when the converse holds. We prove that for cocompact CAT(0) spaces, a homeomorphism of Morse boundaries is induced by a quasi-isometry if and only if the homeomorphism is quasi-mobius and 2-stable.
Let $X$ be a proper CAT(0) space and let $G$ be a cocompact group of isometries of $X$ which acts properly discontinuously. Charney and Sultan constructed a quasi-isometry invariant boundary for proper CAT(0) spaces which they called the contracting boundary. The contracting boundary imitates the Gromov boundary for $δ$-hyperbolic spaces. We will make this comparison more precise by establishing some well known results for the Gromov boundary in the case of the contracting boundary. We show that the dynamics on the contracting boundary is very similar to that of a $δ$-hyperbolic group. In part…
Croke and Kleiner constructed two homeomorphic locally CAT(0) complexes whose universal covers have visual boundaries that are not homeomorphic. We construct two homeomorphic locally CAT(0) complexes so that the visual boundary of one universal cover contains a nonplanar graph, while the visual boundary of the other does not. In contrast, we prove for any two locally CAT(0) metrics on the Croke-Kleiner complex, if a finite graph embeds in the visual boundary of one universal cover, then the graph embeds in the visual boundary of the other.
In this paper, we study CAT(0) spaces with non-locally connected boundary. We give some condition of a CAT(0) space whose boundary is not locally connected.
We introduce a $\mathbb{Z}$-valued cross ratio on Roller boundaries of ${\rm CAT(0)}$ cube complexes. We motivate its relevance by showing that every cross-ratio preserving bijection of Roller boundaries uniquely extends to a cubical isomorphism. Our results are strikingly general and even apply to infinite dimensional, locally infinite cube complexes with trivial automorphism group.
To every Gromov hyperbolic space X one can associate a space at infinity called the Gromov boundary of X. Gromov showed that quasi-isometries of hyperbolic metric spaces induce homeomorphisms on their boundaries, thus giving rise to a well-defined notion of the boundary of a hyperbolic group. Croke and Kleiner showed that the visual boundary of non-positively curved (CAT(0)) groups is not well-defined, since quasi-isometric CAT(0) spaces can have non-homeomorphic boundaries. For any sublinear function $κ$, we consider a subset of the visual boundary called the $κ$-Morse boundary and show that …
In this paper, we investigate the fixed-point set of an element of a CAT(0) group in its boundary. Suppose that a group $G$ acts geometrically on a CAT(0) space $X$. Let $g\in G$ and let $\mathcal{F}_g$ be the fixed-point set of $g$ in the boundary $\partial X$. Then we show that $\mathcal{F}_g=L(Z_g)$, where $Z_g$ is the centralizer of $g$ (i.e. $Z_g=\{v\in G| gv=vg\}$) and $L(Z_g)$ is the limit set of $Z_g$ in $\partial X$. Thus we obtain that $\mathcal{F}_g\neq \emptyset$ if and only if the set $Z_g$ is infinite. We also show that if $g$ is a hyperbolic isometry, then $\mathcal{F}_g=\partia…
We show under weak hypotheses that $\partial X$, the Roller boundary of a finite dimensional CAT(0) cube complex $X$ is the Furstenberg-Poisson boundary of a sufficiently nice random walk on an acting group $Γ$. In particular, we show that if $Γ$ admits a nonelementary proper action on $X$, and $μ$ is a generating probability measure of finite entropy and finite first logarithmic moment, then there is a $μ$-stationary measure on $\partial X$ making it the Furstenberg-Poisson boundary for the $μ$-random walk on $Γ$. We also show that the support is contained in the closure of the regular points…
The Roller boundary is a well-known compactification of a CAT(0) cube complex X. When X is locally finite, essential, irreducible, non-Euclidean and admits a cocompact action by a group G, Nevo-Sageev show that a subset, B(X), of the Roller boundary is the realization of the Poisson boundary and that the action of G on B(X) is minimal and strongly proximal. Additionally, these authors show B(X) satisfies many other desirable dynamical and topological properties. In this article we give several equivalent characterizations for when B(X) is equal to the entire Roller boundary. As an application …
We show under weak hypotheses that the pushforward $\{Z_no\}$ of a random-walk to a CAT(0) cube complex converges to a point on the boundary. We introduce the notion of squeezing points, which allows us to consider the convergence in either the Roller boundary or the visual boundary, with the appropriate hypotheses. This study allows us to show that any nonelementary action necessarily contains regular elements, that is, elements that act as rank-1 hyperbolic isometries in each irreducible factor of the essential core.
