We have developed polynomial-time algorithms to generate terms of the cogrowth series for groups $\mathbb{Z}\wr \mathbb{Z},$ the lamplighter group, $(\mathbb{Z}\wr \mathbb{Z})\wr \mathbb{Z}$ and the Navas-Brin group $B.$ We have also given an improved algorithm for the coefficients of Thompson's group $F,$ giving 32 terms of the cogrowth series. We develop numerical techniques to extract the asymptotics of these various cogrowth series. We present improved rigorous lower bounds on the growth-rate of the cogrowth series for Thompson's group $F$ using the method from \cite{HHR15} applied to our extended series. We also generalise their method by showing that it applies to loops on any locally finite graph. Unfortunately, lower bounds less than 16 do not help in determining amenability.
Again for Thompson's group $F$ we prove that, if the group is amenable, there cannot be a sub-dominant stretched exponential term in the asymptotics\footnote{ }. Yet the numerical data provides compelling evidence for the presence of such a term. This observation suggests a potential path to a proof of non-amenability: If the universality class of the cogrowth sequence can be determined rigorously, it will likely prove non-amenability.
We estimate the asymptotics of the cogrowth coefficients of $F$ to be $$ c_n \sim c \cdot μ^n \cdot κ^{n^σ\log^δ{n}} \cdot n^g,$$ where $μ\approx 15,$ $κ\approx 1/e,$ $σ\approx 1/2,$ $δ\approx 1/2,$ and $g \approx -1.$ The growth constant $μ$ must be 16 for amenability. These two approaches, plus a third based on extrapolating lower bounds, support the conjecture \cite{ERvR15, HHR15} that the group is not amenable.
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…
In this paper, we prove that if two finite groups G and H have isomorphic Burnside rings, then G and H are the same order type groups, and give an example to show that the Burnside rings of the same order type groups are not necessarily isomorphic. This result is related to the Thompson Problem which was raised in 1987.
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…
In this second article, we continue to study classes of groups constructed from a functorial method due to Vaughan Jones. A key observation of the author shows that these groups have remarkable diagrammatic properties that can be used to deduce their properties. Given any group and two of its endomorphisms, we construct a semidirect product. In our first article dedicated to this construction, we classify up to isomorphism all these semidirect products when one of the endomorphisms is trivial and described their automorphism group. In this article we focus on the case where both endomorphisms …
Let $Λ$ be a row-finite and source-free higher rank graph with finitely many vertices. In this paper, we define the Higman-Thompson like group $\Lht$ of the graph C*-algebra $\mathcal{O}_Λ$ to be a special subgroup of the unitary group in $Ø_Λ$. It is shown that $\Lht$ is closely related to the topological full groups of the groupoid associated with $Λ$. Some properties of $\Lht$ are also investigated. We show that its commutator group $\DLht$ is simple and that $\DLht$ has only one nontrivial uniformly recurrent subgroup if $Λ$ is aperiodic and strongly connected. Furthermore, if $Λ$ is singl…
We study skew-amenable topological groups, i.e., those admitting a left-invariant mean on the space of bounded real-valued functions left-uniformly continuous in the sense of Bourbaki. We prove characterizations of skew-amenability for topological groups of isometries and automorphisms, clarify the connection with extensive amenability of group actions, establish a Følner-type characterization, and discuss closure properties of the class of skew-amenable topological groups. Moreover, we isolate a dynamical sufficient condition for skew-amenability and provide several concrete variations of thi…
Through the glasses of didactic reduction: We consider a (periodic) tessellation $Δ$ of either Euclidean or hyperbolic $n$-space $M$. By a piecewise isometric rearrangement of $Δ$ we mean the process of cutting $M$ along corank-1 tile-faces into finitely many convex polyhedral pieces, and rearranging the pieces to a new tight covering of the tessellation $Δ$. Such a rearrangement defines a permutation of the (centers of the) tiles of $Δ$, and we are interested in the group $PI(Δ)$ of all piecewise isometric rearrangements of $Δ$. In this paper we offer: a) An illustration of piecewise isometri…
