We study the monoid generalization M_{k,1} of the Thompson-Higman groups, and we characterize the R- and the L-preorder of M_{k,1}. Although M_{k,1} has only one non-zero J-class and k-1 non-zero D-classes, the R- and the L-preorder are complicated; in particular, <_R is dense (even within an L-class), and <_L is dense (even within an R-class).
We study the computational complexity of the R- and the L-preorder. When inputs are given by words over a finite generating set of M_{k,1}, the R- and the L-preorder decision problems are in P. The main result of the paper is that over a "circuit-like" generating set, the R-preorder decision problem of M_{k,1} is Pi_2^P-complete, whereas the L-preorder decision problem is coNP-complete. We also prove related results about circuits: For combinational circuits, the surjectiveness problem is Pi_2^P-complete, whereas the injectiveness problem is coNP-complete.
We investigate stabilizers of finite sets of rational points in Cantor space for the Higman-Thompson groups $V_{n,r}$. We prove that the pointwise stabilizer is an iterated ascending HNN extension of $V_{n,q}$ for any $q\geq 1$. We also prove that the commutator subgroup of the pointwise stabilizer is simple, and we compute the abelianization. Finally, for each $n$ we classify such pointwise stabilizers up to isomorphism.
We generalize the Brin-Higman-Thompson groups $n G_{k,1}$ to monoids $n M_{k,1}$, for $n \ge 1$ and $k \ge 2$, by replacing bijections by partial functions. The monoid $n M_{k,1}$ has $n G_{k,1}$ as its group of units, and is congruence-simple. Moreover, $n M_{k,1}$ is finitely generated, and for $n \ge 2$ its word problem is {\sf coNP}-complete. We also present new results about higher-dimensional joinless codes.
We show that the word problem of the Brin-Higman-Thompson group $n G_{k,1}$ is {\sf coNP}-complete for all $n \ge 2$ and all $k \ge 2$. For this we prove that $n G_{k,1}$ is finitely generated, and that $n G_{k,1}$ contains a subgroup of $2 G_{2,1}$ that can represent bijective circuits. We also show that for all $n \ge 1$ and $k \ge 2$: \ If $\,K = 1 + (k-1)\,N\,$ for some $N \ge 1$, then $n G_{K,1} \le n G_{k,1}$. In particular, $n G_{K,1} \le n G_{2,1}$ for all $K \ge 2$.
Every representation of the Cuntz algebra $\mathcal{O}_n$ leads to a unitary representation of the Higman-Thompson group $V_n$. We consider the family $\{π_x\}_{x\in [0,1[}$ of permutative representations of $\mathcal{O}_n$ that arise from the interval map $f(x)=nx$ (mod 1) acting on the Hilbert space that underlies each orbit, and then study the unitary equivalence and the irreducibility of the corresponding family $\{ρ_x\}_{x\in [0,1[}$ of representations of Higman-Thompson group $V_n$, showing that that these representations are indeed irreducible and moreover $ρ_x$ and $ρ_y$ are equivalent…
Let $1 \le r < n$ be integers. We give a proof that the group $\mathop{\mathrm{Aut}}({X_{n}^{\mathbb{N}}, σ_{n}})$ of automorphisms of the one-sided shift on $n$ letters embeds naturally as a subgroup $\mathcal{H}_{n}$ of the outer automorphism group $\mathop{\mathrm{Out}}({G_{n,r}})$ of the Higman-Thompson group $G_{n,r}$. From this, we can represent the elements of $\mathop{\mathrm{Aut}}({X_{n}^{\mathbb{N}}, σ_{n}})$ by finite state non-initial transducers admitting a very strong synchronizing condition. Let $H \in \mathcal{H}_{n}$ and write $|H|$ for the number of states of the minimal tran…
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 revisit our construction of the Thompson groups from the polycyclic inverse monoids in the light of new research. Specifically, we prove that the Thompson group $G_{n,1}$ is the group of units of a Boolean inverse monoid $C_{n}$ called the Cuntz inverse monoid. This inverse monoid is proved to be the tight completion of the polycyclic inverse monoid $P_{n}$. The étale topological groupoid associated with $C_{n}$ under non-commutative Stone duality is the usual groupoid associated with the corresponding Cuntz $C^{\ast}$-algebra. We then show that the group $G_{n,1}$ is also the group of auto…
In this article, we further explore the nature of a connection between the groups of automorphisms of full shift spaces and the groups of outer automorphisms of the Higman--Thompson groups $\{G_{n,r}\}$. We show that the quotient of the group of automorphisms of the (two-sided) shift dynamical system $\mathrm{Aut}(X_n^{\mathbb{N}}, σ_{n})$ by its centre embeds as a particular subgroup $\mathcal{L}_{n}$ of the outer automorphism group $\mathop{\mathrm{Out}}(G_{n,n-1})$ of $G_{n,n-1}$. It follows by a result of Ryan that we have the following central extension: $$\langle σ_{n}\rangle \hookrighta…
Monoid generalizations of the Richard Thompson groups
Richard Thompson和Graham Higman的群G_{k,1}可以自然地推广到幺半群，我们称之为M_{k,1}，以及逆幺半群，称为Inv_{k,1} ;这可以通过简单地将双射推广到部分函数或部分内射函数来完成。幺半群M_{k,1}与电路复杂性有关（在另一篇论文中研究）。本文证明了M_{k,1}和Inv_{k,1}对所有k都是同余单的.它们的格林关系J和D被刻画为：M_{k,1}和Inv_{k,1}是J-0-单的，并且它们有k-1个非零的D-类.它们是Cuntz代数O_k的乘法部分的子幺半群。它们都是有限生成的，它们在任意有限生成集上的字问题都在P中，它们的字问题在某些无限生成集上是coNP-完全的.本版本的变化：第4节已被彻底修订，错误已被纠正;但是，第4节的主要结果没有变化。除了定理2.3的证明是不完整的外，第1、2和3节没有改变;完整的证明发表在参考文献[6]的附录中，这里也给出了。
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…
We introduce a new distributional invariance principle, called `partial spreadability', which emerges from the representation theory of the Thompson monoid $F^+$ in noncommutative probability spaces. We show that a partially spreadable sequence of noncommutative random variables is adapted to a local Markov filtration. Conversely we show that a large class of noncommutative stationary Markov sequences provides representations of the Thompson monoid $F^+$. In the particular case of a classical probability space, we arrive at a de Finetti theorem for stationary Markov sequences with values in a …
The Thompson group $V$, as well as the Brin-Thompson group $2V$, is finitely generated and can be defined as a monoid acting on bitstrings, respectively pairs of bitstrings. Therefore evaluation problems can be defined for $V$ and $2V$. We show that these evaluation problems reduce to the corresponding word problems, and that in general, these evaluation problems are actually equivalent to the word problems. The long-input version of the evaluation problem is deterministic context-free and reverse deterministic context-free for $V,$ and P-complete for $2V$.
Groups with Context-Free Co-Word Problem and Embeddings into Thompson's Group $V$
设$G$是一个群生成的群，$Σ$是一个生成$G$为幺半群的有限子集。\emph{word problem of $G$ with respect to $Σ$}由自由幺半群$Σ^{\ast}$中所有与$G$中的恒等式相等的字组成。\emph{co-word problem of $G$ with respect to $Σ$}是字问题在$Σ^{\ast}$中的补充。我们说一个群$G$是\emph{co$\mathcal{CF}$}，如果它关于某个（等价地，任何）有限生成集$Σ$的共字问题是上下文无关语言。我们对每个有限群G描述了一个广义Thompson群$V_{(G, θ)}$和同态$θ$：$G \rightarrow G$ .我们的群体是使用Witzel和Zaremsky介绍的克隆系统构建的。通过构造一个下推自动机，证明了对于任意同态$θ$和有限群G，$V_{(G, θ)}$是coPDMATHABMV，并证明了$V_{(G, θ)}$的co-字问题是我们的自动机所接受的语言的循环移位.一个版本的猜想，由于Lehnert说，一个组有上下文无关的共同字问题正是如果它是一个生成子群的V.组$V_{(G,θ)}$，其中$θ$不是单位同态似乎没有明显的嵌入到V，因此可能被视为反例的猜想。由Bleak和Salazar-Diaz引入的$V$的示范子群可以用来构造某些环积和合并自由积到$V$中的…
