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 of an elementary amenable subgroup up to class (height) omega squared, where omega is the first infinite ordinal.
Richard Thompson's group F is the group of piecewise linear homeomorphisms of the unit interval with a finite number of break points, all at dyadic rational numbers (their denominators are powers of 2) and with slopes which are powers of 2. A discrete group G is amenable if there exists a finitely-additive probability measure on G which is invariant under left translations and is defined on all subsets of G. The amenability question for F is a well known open problem. In this paper we prove that group F is not amenable.
This paper demonstrates the uniformly finite homology developed by Block and Weinberger and its relationship to amenable spaces via applications to the Cayley graph of Thompson's Group F. In particular, a certain class of subgraph of F is shown to be non-amenable. This shows that if F is amenable, these subsets (which include every finitely generated submonoid of the positive monoid of F) must necessarily have measure zero.
We investigate the notion of relatively amenable topological action and show that the action of Thompson's group $T$ on $S^1$ is relatively amenable with respect to Thompson's group $F$. We use this to conclude that $F$ is exact if and only if $T$ is exact. Moreover, we prove that the groupoid of germs of the action of $T$ on $S^1$ is Borel amenable.
In this short note, we show that R. Thompson's group $F$ admits a normalish amenable subgroup, and that the standard copy of $F$ in R. Thompson's group $T$ is normalish in $T$. We further conjecture that if $F$ is non-amenable, then $T$ does not admit a normalish amenable subgroup, and therefore that the reduced $C^*$ algebra of $T$ is in fact simple in that case.
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 descripti…
In this paper we prove that the Thompson groups $T$ and $V$ are not inner amenable. In particular, their group von Neumann algebras do not have property $Γ$. Moreover, we prove that if the reduced group $C^\ast$-algebra of $T$ is simple, then the Thompson group $F$ is non-amenable. Furthermore, we give a few new equivalent characterizations of amenability of $F$.
We prove that any finitely generated elementary amenable group of zero (algebraic) entropy contains a nilpotent subgroup of finite index or, equivalently, any finitely generated elementary amenable group of exponential growth is of uniformly exponential growth. We also show that 0 is an accumulation point of the set of entropies of elementary amenable groups.
There are several natural families of groups acting on rooted trees for which every member is known to be amenable. It is, however, unclear what the elementary amenable members of these families look like. Towards clarifying this situation, we here study elementary amenable bounded automata groups. We are able to isolate the elementary amenable bounded automata groups in three natural subclasses of bounded automata groups. In particular, we show that iterated monodromy groups of post-critically finite polynomials are either virtually abelian or not elementary amenable.
We consider groups of automorphisms of locally finite trees, and give conditions on its subgroups that imply that they are not elementary amenable. This covers all known examples of groups that are not elementary amenable and act on the trees: groups of intermediate growths and Basilica group, by giving a more straightforward proof. Moreover, we deduce that all finitely generated branch groups are not elementary amenable, which was conjectured by Grigorchuk.
We describe the results of some computational explorations in Thompson's group F. We describe experiments to estimate the cogrowth of F with respect to its standard finite generating set, designed to address the subtle and difficult question whether or not Thompson's group is amenable. We also describe experiments to estimate the exponential growth rate of F and the rate of escape of symmetric random walks with respect to the standard generating set.
In this paper we prove that the general version, F(N) of the Thompson group is inner amenable. As a consequence we generalize a result of P.Jolissaint. To do so, we prove first that F(N) together with a normal subgroup are i.c.c (infinite conjugacy classes) groups. Then, we investigate the relative McDuff property out of which we extract property $Γ$ for the group von Neumann algebras involved. By a result of E.G.Effros, F(N) follows inner amenable.
A group $G$ is called mixed identity-free if for every $n \in \mathbb{N}$ and every $w \in G \ast F_n$ there exists a homomorphism $φ: G \ast F_n \rightarrow G$ such that $φ$ is the identity on $G$ and $φ(w)$ is nontrivial. In this paper, we make a modification to the construction of elementary amenable lacunary hyperbolic groups given by Ol'shanskii, Osin, and Sapir to produce finitely generated elementary amenable groups which are mixed identity-free. As a byproduct of this construction, we also obtain locally finite $p$-groups which are mixed identity-free.
The paper has three parts. It is conjectured that for every elementary amenable group G and every non-zero commutative ring k, the homological dimension of G over k is equal to the Hirsch length of G whenever G has no k-torsion. In Part I this is proved for several classes, including the abelian-by-polycyclic groups. In Part II it is shown that elementary amenable groups of homological dimension one are filtered colimits of systems of groups of cohomological dimension one. Part III is devoted to the deeper study of cohomological dimension with particular emphasis on the nilpotent-by-polycyclic…
We demonstrate the existence of a family of finitely generated subgroups of Richard Thompson's group $F$ which is strictly well-ordered by the embeddability relation in type $ε_0 +1$. All except the maximum element of this family (which is $F$ itself) are elementary amenable groups. In fact we also obtain, for each $α< ε_0$, a finitely generated elementary amenable subgroup of $F$ whose EA-class is $α+ 2$. These groups all have simple, explicit descriptions and can be viewed as a natural continuation of the progression which starts with $\mathbf{Z} + \mathbf{Z}$, $\mathbf{Z} \wr \mathbf{Z}$, a…
The Schreier graphs of Thompson's group F with respect to the stabilizer of 1/2 and generators x_0 and x_1, and of its unitary representation in L_2([0,1]) induced by the standard action on the interval [0,1] are explicitly described. The coamenability of the stabilizers of any finite set of dyadic rational numbers is established. The induced subgraph of the right Cayley graph of the positive monoid of F containing all the vertices of the form x_nv, where n>=0 and v is any word over the alphabet {x_0, x_1}, is constructed. It is proved that the latter graph is non-amenable.
Given a set S equipped with a binary operation (we call this a "bracket algebra") one may ask to what extent the binary operation satisfies some of the consequences of the associative law even when it is not actually associative? We define a subgroup Assoc(S) of Thompson's Group F for each bracket algebra S, and we interpret the size of Assoc(S) as determining the amount of associativity in S - the larger Assoc(S) is, the more associativity holds in S. When S is actually associative, Assoc(S) = F; that is the trivial case. In general, it turns out that only certain subgroups of F can occur as …
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…
It is a long standing open problem whether the Thompson group $F$ is an amenable group. In this paper we show that if $A$, $B$, $C$ denote the standard generators of Thompson group $T$ and $D:=C B A^{-1}$ then $$\sqrt2+\sqrt3\,<\,\frac1{\sqrt{12}}||(I+C+C^2)(I+D+D^2+D^3)||\,\le\, 2+\sqrt2.$$ Moreover, the upper bound is attained if the Thompson group $F$ is amenable. Here, the norm of an element in the group ring $\mathbb{C} T$ is computed in $B(\ell^2(T))$ via the regular representation of $T$. Using the "cyclic reduced" numbers $τ(((C+C^2)(D+D^2+D^3))^n)$, $n\in\mathbb{N}$, and some methods …
We show that the group C*-algebra of any elementary amenable group is quasidiagonal. This is an offspring of recent progress in the classification theory of nuclear C*-algebras.
We show that elementary amenable groups, which have a bound on the orders of their finite subgroups, admit a finite dimensional model for the classifying space with virtually cyclic isotropy.
Let $R=K[G]$ be a group ring of a group $G$ over a field $K$. It is known that if $G$ is amenable then $R$ satisfies the Ore condition: for any $a,b\in R$ there exist $u,v\in R$ such that $au=bv$, where $u\ne0$ or $v\ne0$. It is also true for amenable groups that a non-zero solution exists for any finite system of linear equations over $R$, where the number of unknowns exceeds the number of equations. Recently Bartholdi proved the converse. As a consequence of this theorem, Kielak proved that R.\,Thompson's group $F$ is amenable if and only if it satisfies the Ore condition. The amenability pr…
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 first consider three well-known chain conditions in the space of marked groups: the minimal condition on centralizers, the maximal condition on subgroups, and the maximal condition on normal subgroups. For each condition, we produce a characterization in terms of well-founded descriptive-set-theoretic trees. Using these characterizations, we demonstrate that the sets given by these conditions are co-analytic and not Borel in the space of marked groups. We then adapt our techniques to show elementary amenable marked groups may be characterized by well-founded descriptive-set-theoretic trees,…
We show that for a large class $\mathcal{C}$ of finitely generated groups of orientation preserving homeomorphisms of the real line, the following holds: Given a group $G$ of rank $k$ in $\mathcal{C}$, there is a sequence of $k$-markings $(G, S_n), n\in \mathbf{N}$ whose limit in the space of marked groups is the free group of rank $k$ with the standard marking. The class we consider consists of groups that admit actions satisfying mild dynamical conditions and a certain "self-similarity" type hypothesis. Examples include Thompson's group $F$, Higman-Thompson groups, Stein-Thompson groups, var…
This paper examines Shavgulidze's postings to the ArXiv and publications which concern his argument that Thompson's group F is amenable. In particular I list specific places where there are errors in each of the postings and publications and give a public clarification of some of the speculation around these papers.
We show that the generation problem in Thompson group $F$ is decidable, i.e., there is an algorithm which decides if a finite set of elements of $F$ generates the whole $F$. The algorithm makes use of the Stallings $2$-core of subgroups of $F$, which can be defined in an analogue way to the Stallings core of subgroups of a finitely generated free group. Further study of the Stallings $2$-core of subgroups of $F$ provides a solution to another algorithmic problem in $F$. Namely, given a finitely generated subgroup $H$ of $F$, it is decidable if $H$ acts transitively on the set of finite dyadic …
In a recent paper Uffe Haagerup and Kristian Knudsen Olesen show that for Richard Thompson's group $T$, if there exists a finite set $H$ which can be decomposed as disjoint union of sets $H_1$ and $H_2$ with $\sum_{g\in H_1}π(g)=\sum_{h\in H_2}π(h)$ and such that the closed ideal generated by $\sum_{g\in H_1}λ(g)-\sum_{h\in H_2}λ(h)$ coincides with $C^*_λ(T)$, then the Richard Thompson group $F$ is not amenable. In particular, if $C_λ^*(T)$ is simple then $F$ is not amenable. Here we prove the converse, namely, if $F$ is not amenable then we can find two sets $H_1$ and $H_2$ with the above pro…
We show that locally solvable subgroups of PLo(I) are countable. Then for each countable ordered set, we construct a locally solvable subgroup of Thompson's Group F. We develop machinery for understanding embeddings from solvable subgroups into solvable subgroups. Finally, we apply this machinery to show the ordered sets used in our construction are invariant under isomorphisms between the groups constructed. Therefore, we effectively distinguish the groups and provide uncountably many non-isomorphic locally solvable, hence elementary amenable, subgroups of Thompson's Group F.
Schreier graphs of the actions of Thompson's group $F$ on the orbits of all points of the unit interval and of the Cantor set with respect to the standard generating set $\{x_0,x_1\}$ are explicitly constructed. The closure of the space of pointed Schreier graphs of the action of $F$ on the orbits of dyadic rational numbers and corresponding Schreier dynamical system are described. In particular, we answer the question of Grigorchuk on the Cantor-Bendixson rank of the underlying space of the Schreier dynamical system in the context of $F$. As applications we prove that the pointed Schreier gra…
In this short article, we prove that any automorphism of the R. Thompson's group $F$ has infinitely many twisted conjugacy classes. The result follows from the work of Matthew Brin, together with a standard facts on R. Thompson's group $F$, and elementary properties of the Reidemeister numbers.
We prove a variety of results about the group von Neumann algebras associated to Thompson-like groups arising from so called $d$-ary cloning systems. Cloning systems are a framework developed by Witzel and the second author, with a $d$-ary version subsequently developed by Skipper and the second author, which can be used to construct generalizations of the classical Thompson's groups $F$, $T$, and $V$. Given a family of groups $(G_n)_{n\in\mathbb{N}}$ with a $d$-ary cloning system, we get a Thompson-like group $\mathscr{T}_d(G_*)$, and in this paper we find some mild, natural conditions under …
We show that the asymptotic dimension of box spaces behaves (sub)additively with respect to extensions of groups. As a result, we obtain that for an elementary amenable group, the asymptotic dimension of any of its box spaces is bounded above by its Hirsch length. This bound is shown to be an equality for a large subclass of groups including all virtually polycyclic groups.
We show that every group $H$ of at most exponential growth with respect to some left invariant metric admits a bi-Lipschitz embedding into a finitely generated group $G$ such that $G$ is amenable (respectively, solvable, satisfies a non-trivial identity, elementary amenable, of finite decomposition complexity, etc.) whenever $H$ is. We also discuss some applications to compression functions of Lipschitz embeddings into uniformly convex Banach spaces, Følner functions, and elementary classes of amenable groups.
The purpose of this article is prove that Thompson's group F is amenable. The methods developed will then be used to prove a generalization of Hindman's theorem for the free nonassociative binary system on one generator.
We study the uniformly recurrent subgroups of groups acting by homeomorphisms on a topological space. We prove a general result relating uniformly recurrent subgroups to rigid stabilizers of the action, and deduce a $C^*$-simplicity criterion based on the non-amenability of rigid stabilizers. As an application, we show that Thompson's group $V$ is $C^\ast$-simple, as well as groups of piecewise projective homeomorphisms of the real line. This provides examples of finitely presented $C^\ast$-simple groups without free subgroups. We prove that a branch group is either amenable or $C^\ast$-simple…
In this paper we prove that the homological dimension of an elementary amenable group over an arbitrary commutative coefficient ring is either infinite or equal to the Hirsch length of the group. Established theory gives simple group theoretical criteria for finiteness of homological dimension and so we can infer complete information about this invariant for elementary amenable groups. Stammbach proved the special case of solvable groups over coefficient fields of characteristic zero in an important paper dating from 1970.
We study subgroups $H_U$ of the R. Thompson group $F$ which are stabilizers of finite sets $U$ of numbers in the interval $(0,1)$. We describe the algebraic structure of $H_U$ and prove that the stabilizer $H_U$ is finitely generated if and only if $U$ consists of rational numbers. We also show that such subgroups are isomorphic surprisingly often. In particular, we prove that if finite sets $U\subset [0,1]$ and $V\subset [0,1]$ consist of rational numbers which are not finite binary fractions, and $|U|=|V|$, then the stabilizers of $U$ and $V$ are isomorphic. In fact these subgroups are conju…
We study some properties of the Cayley graph of the R.Thompson's group F in generators $x_0$, $x_1$. We show that the density of this graph, that is, the least upper bound of the average vertex degree of its finite subgraphs is at least 3. It is known that a 2-generated group is not amenable if and only if the density of the corresponding Cayley graph is strictly less than 4. It is well known this is also equivalent to the existence of a doubling function on the Cayley graph. This means there exists a mapping from the set of vertices into itself such that for some constant $K>0$, each vertex m…
We study actions of groups by homeomorphisms on $\mathbf{R}$ (or an interval) that are minimal, have solvable germs at $\pm \infty$ and contain a pair of elements of a certain type. We call such actions coherent. We establish that such an action is rigid, i.e. any two such actions of the same group are topologically conjugate. We also establish that the underlying group is always non elementary amenable, but satisfies that every proper quotient is solvable. As a first application, we demonstrate that any coherent group action $G<Homeo^+(\mathbf{R})$ that produces a nonamenable equivalence rela…
We prove that the approximation conjecture of Luck holds for all amenable groups in the complex group algebra case. This result was previously proved by Dodziuk, Linnell, Mathai, Schick and Yates under the assumption that the group is torsion-free elementary amenable.
We describe elementary examples of finitely presented sofic groups which are not residually amenable (and thus not initially subamenable or LEA, for short). We ask if an amalgam of two amenable groups over a finite subgroup is residually amenable and answer this positively for some special cases, including countable locally finite groups, residually nilpotent groups and others.
We give a complete characterization of the locally compact groups that are non-elementary Gromov-hyperbolic and amenable. They coincide with the class of mapping tori of discrete or continuous one-parameter groups of compacting automorphisms. We moreover give a description of all Gromov-hyperbolic locally compact groups with a cocompact amenable subgroup: modulo a compact normal subgroup, these turn out to be either rank one simple Lie groups, or automorphism groups of semi-regular trees acting doubly transitively on the set of ends. As an application, we show that the class of hyperbolic loca…
We provide equivalent conditions for Liouville property of actions of groups. As an application, we show that there is a Liouville measure for the action of the Thompson group $F$ on dyadic rationals. This result should be compared with a recent result of Kaimanovich, where he shows that the action of the Thompson group F on dyadic rationals is not Liouville for all finitely supported measures. As another application we show that there is a Liouville measure for lamplighter actions. This gives more examples of non-amenable Liouville actions.
Let $R=K[G]$ be a group ring of a group $G$ over a field $K$. The Ore condition says that for any $a,b\in R$ there exist $u,v\in R$ such that $au=bv$, where $u\ne0$ or $v\ne0$. It always holds whenever $G$ is amenable. Recently it was shown that for R.\,Thompson's group $F$ the converse is also true. So the famous amenability problem for $F$ is equivalent to the question on the Ore condition for the group ring of the same group. It is easy to see that the problem on the Ore condition for $K[F]$ is equivalent to the same property for the monoid ring $K[M]$, where $M$ is the monoid of positive e…
We introduce the notion of groups of polytope class and show that torsion-free amenable groups satisfying the Atiyah Conjecture possess this property. A direct consequence is the homotopy invariance of the $L^2$-torsion polytope among $G$-CW-complexes for these groups. As another application we prove that the $L^2$-torsion polytope of an amenable group vanishes provided that it contains a non-abelian elementary amenable normal subgroup.
Let $F$ denote the Thompson group with standard generators $A=x_0$, $B=x_1$. It is a long standing open problem whether $F$ is an amenable group. By a result of Kesten from 1959, amenability of $F$ is equivalent to $$(i)\qquad ||I+A+B||=3$$ and to $$(ii)\qquad ||A+A^{-1}+B+B^{-1}||=4,$$ where in both cases the norm of an element in the group ring $\mathbb{C} F$ is computed in $B(\ell^2(F))$ via the regular representation of $F$. By extensive numerical computations, we obtain precise lower bounds for the norms in $(i)$ and $(ii)$, as well as good estimates of the spectral distributions of $(I+A…
This article is dedicated to the study of asymptotically rigid mapping class groups of infinitely-punctured surfaces obtained by thickening planar trees. Such groups include the braided Ptolemy-Thompson groups $T^\sharp,T^\ast$ introduced by Funar and Kapoudjian, and the braided Houghton groups $\mathrm{br}H_n$ introduced by Degenhardt. We present an elementary construction of a contractible cube complex, on which these groups act with cube-stabilisers isomorphic to finite extensions of braid groups. As an application, we prove Funar-Kapoudjian's and Degenhardt's conjectures by showing that $T…
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）
P-E. Caprace and N. Monod isolate the class $\mathscr{X}$ of locally compact groups for which relatively amenable closed subgroups are amenable. It is unknown if $\mathscr{X}$ is closed under group extension. In this note, we exhibit a large, group extension stable subclass of $\mathscr{X}$, which suggests $\mathscr{X}$ is indeed closed under group extension. Along the way, we produce generalizations of the class of elementary groups and obtain information on groups outside $\mathscr{X}$.
The Arithmetic is interpreted in all the groups of Richard Thompson and Graham Higman, as well as in other groups of piecewise affine permutations of an interval which generalize the groups of Thompson and Higman. In particular, the elementary theories of all these groups are undecidable. Moreover, Thompson's group $F$ and some of its generalizations interpret the Arithmetic without parameters.
