We study when the group $\mathbb Z^n\rtimes_A\mathbb Z$ is arithmetic where $A\in GL_n(\mathbb Z)$ is hyperbolic and semisimple. We begin by giving a characterization of arithmeticity phrased in the language of algebraic tori, building on work of Grunewald-Platonov. We use this to prove several more concrete results that relate the arithmeticity of $\mathbb Z^n\rtimes_A\mathbb Z$ to the reducibility properties of the characteristic polynomial of $A$. Our tools include algebraic tori, representation theory of finite groups, Galois theory, and the inverse Galois problem.
We initiate a new, computational approach to a classical problem: certifying non-freeness of ($2$-generator, parabolic) Möbius subgroups of $\mathrm{SL}(2,\mathbb{Q})$. The main tools used are algorithms for Zariski dense groups and algorithms to compute a presentation of $\mathrm{SL}(2, R)$ for a localization $R= \mathbb{Z}[\frac{1}{b}]$ of $\mathbb{Z}$. We prove that a Möbius subgroup $G$ is not free by showing that it has finite index in the relevant $\mathrm{SL}(2, R)$. Further information about the structure of $G$ is obtained; for example, we compute the minimal subgroup of finite index …
Let $G$ and $N$ be finite groups of order $2n$ where $n$ is odd. We say the pair $(G,N)$ is Hopf-Galois realizable if $G$ is a regular subgroup of $\h(N)=N\rtimes\au(N)$. In this article we give necessary conditions on $G$ (similarly $N$) when $N$ (similarly $G$) is a group of the form $\mathbb{Z}_n\rtimes\mathbb{Z}_2$. Further we show that this condition is also sufficient if radical of $n$ is a Burnside number. This classifies all the skew braces which has the additive group (or the multiplicative group) to be isomorphic to $\mathbb{Z}_n\rtimes\mathbb{Z}_2$, in this case.
We give here the exact maximal subgroup growth of two classes of polycyclic groups. Let $G_k = \langle x_1, x_2, ..., x_k \mid x_ix_jx_i^{-1}x_j \text{ for all } i < j \rangle$. So $G_k = \mathbb{Z} \rtimes (\mathbb{Z} \rtimes (\mathbb{Z} \rtimes ... \rtimes \mathbb{Z})$. Then for all $k \geq 2$, we calculate $m_n(G_k)$, the number of maximal subgroups of $G_k$ of index $n$, exactly. Also, for infinitely many groups $H_k$ of the form $\mathbb{Z}^2 \rtimes G_2$, we calculate $m_n(H_k)$ exactly.
We study invariant random subgroups (IRSs) of semidirect products $G = A \rtimes Γ$. In particular, we characterize all IRSs of parabolic subgroups of $\mathrm{SL}_d(\mathbb{R})$, and show that all ergodic IRSs of $\mathbb{R}^d \rtimes \mathrm{SL}_d(\mathbb{R})$ are either of the form $\mathbb{R}^d \rtimes K$ for some IRS of $\mathrm{SL}_d(\mathbb{R})$, or are induced from IRSs of $Λ\rtimes \mathrm{SL}(Λ)$, where $Λ< \mathbb{R}^d$ is a lattice.
In this note we study the Reidemeister spectrum for metabelian groups of the form ${\mathbb Q}^n\rtimes \mathbb Z$ and ${\mathbb Z[1/p]}^n\rtimes \mathbb Z$. Particular attention is given to the $R_{\infty}$ property of a subfamily of these groups. We also define a Nielsen spectrum of a space and discuss some examples.
Given a short exact sequence of groups with certain conditions, $1\to F\to G\to H\to 1$, we prove that $G$ has solvable conjugacy problem if and only if the corresponding action subgroup $A\leqslant Aut(F)$ is orbit decidable. From this, we deduce that the conjugacy problem is solvable, among others, for all groups of the form $\mathbb{Z}^2\rtimes F_m$, $F_2\rtimes F_m$, $F_n \rtimes \mathbb{Z}$, and $\mathbb{Z}^n \rtimes_A F_m$ with virtually solvable action group $A\leqslant GL_n(\mathbb{Z})$. Also, we give an easy way of constructing groups of the form $\mathbb{Z}^4\rtimes F_n$ and $F_3\rti…
Let $K/F$ be a finite Galois extension of fields with $Gal(K/F)=Γ$. In an earlier work of Timothy Kohl, the author enumerated dihedral Hopf-Galois structures acting on dihedral extensions. Dihedral group is one particular example of semidirect product of $\mathbb{Z}_n$ and $\mathbb{Z}_2$. In this article we count the number of Hopf-Galois structures with Galois group $Γ$ of type $G$, where $Γ,G$ are groups of the form $\mathbb{Z}_N\rtimes_ϕ\mathbb{Z}_2$ when $N$ is odd with radical of $N$ being a Burnside number. As an application we also find the corresponding number of skew braces.
By arithmeticity and superrigidity, a commensurability class of lattices in a higher rank Lie group is defined by a unique algebraic group over a unique number subfield of $\mathbb{R}$ or $\mathbb{C}$. We prove an adelic version of superrigidity which implies that two such commensurability classes define the same profinite commensurability class if and only if the algebraic groups are adelically isomorphic. We discuss noteworthy consequences on profinite rigidity questions.
It is proved that the semigroups $\mathrm{\mathbf{End}}(\boldsymbol{B}_ω)$ and $\mathrm{\mathbf{End}}(\boldsymbol{B}_{\mathbb{Z}})$ of the endomorphisms of the bicyclic semigroup $\boldsymbol{B}_ω$ and the endomorphisms of the extended bicyclic semigroup $\boldsymbol{B}_{\mathbb{Z}}$ are isomorphic to the semidirect products $(ω,+)\rtimes_φ(ω,*)$ and $\mathbb{Z}(+)\rtimes_φ(ω,*)$, respectively.
We show that the semidirect product of a group $C$ by $A*_D B$ is isomorphic to the free product of $A\rtimes C$ and $B\rtimes C$ amalgamated at $D\rtimes C$, where $A$, $B$ and $C$ are arbitrary groups. Moreover, we apply this theorem to prove that any group $G$ that acts without inversion on a tree $T$ that possesses a segment $Γ$ for its quotient graph, such that, if the stabilizers of the vertex set $\{\,P,Q\,\}$ and edge $y$ of a lift of $\, Γ$ in $T$ are of the form $G_{P}\!\rtimes H$, $G_{Q}\!\rtimes H$ and $G_{y}\! \rtimes H$, then $G$ is isomorphic to the semidirect product of $H$ by …
In this paper, the $H\mathbb Z$-length of different groups is studied. By definition, this is the length of $H\mathbb Z$-localization tower or the length of transfinite lower central series of $H\mathbb Z$-localization. It is proved that, for a free noncyclic group, its $H\mathbb Z$-length is $\geq ω+2$. For a large class of $\mathbb Z[C]$-modules $M,$ where $C$ is an infinite cyclic group, it is proved that the $H\mathbb Z$-length of the semi-direct product $M\rtimes C$ is $\leq ω+1$ and its $H\mathbb Z$-localization can be described as a central extension of its pro-nilpotent completion. In …
We consider a block $B$ of a finite group with defect group $D \cong (C_{2^m})^n$ and inertial quotient $\mathbb{E}$ containing a Singer cycle (an element of order $2^n-1$). This implies $\mathbb{E} = E \rtimes F$, where $E \cong C_{2^n-1}$, $F \leq C_n$, and $E$ acts transitively on the elements in $D$ of order $2$, and freely on $D \backslash \{1\}$. We classify the basic Morita equivalence classes of $B$ over a complete discrete valuation ring $\mathcal{O}$: when $m=1$, $B$ is basic Morita equivalent to the principal block of one of $SL_2(2^n) \rtimes F$, $D \rtimes \mathbb{E}$, or $J_1$ (w…
We show that Cantor minimal $\mathbb{Z}\rtimes\mathbb{Z}_2$-systems and essentially free amenable odometers are almost finite. We also compute the homology groups of Cantor minimal $\mathbb{Z}\rtimes\mathbb{Z}_2$-systems and show that the associated transformation groupoids satisfy the HK conjecture if and only if the action is free.
In this paper we establish lower and upper bounds for the cardinality of the profinite genus of the fundamental group $π_{1}(M_A)\cong (\mathbb{Z} \times \mathbb{Z})\rtimes_{A}\mathbb{Z}$ of a torus bundle $M_{A}$ in terms of the number of ideal classes of the order $\mathbb{Z}[λ]$, where $λ$ is an eigenvalue of the matrix $A$ in $\mathrm{GL}_{2}(\mathbb{Z})$.
We present necessary and sufficient conditions for the existence of a splitting over $\mathbb{Z}$ of the mapping torus $M_ϕ=F\rtimes \mathbb{Z}$ of a free group automorphism $ϕ$.
In this paper, we investigate finitely generated groups of isometries of CAT(0) spaces containing some central hyperbolic isometry, and study CAT(0) groups. We show that every CAT(0) group $Γ$ has the semi-direct product structure $Γ=(\cdots(((Γ'\rtimes\langleδ_{n}\rangle)\rtimes\langleδ_{n-1}\rangle)\rtimes\langleδ_{n-2}\rangle)\cdots)\rtimes\langleδ_{1}\rangle$ where $Γ'$ is a CAT(0) group with finite center and $δ_i\in Γ$ for $i=1,\dots,n$, and $Γ$ contains a finite-index subgroup $Γ'\times A$ where $A$ is isomorphic to ${\mathbb{Z}}^n$. We introduce some examples and remarks. Also we provi…
In this note we give more easy and short proof of a statement previously proved by P. Kahn that the automorphism group of the discrete Heisenberg group ${\rm Heis}(3, \mathbb{Z}) $ is isomorphic to the group $ (\mathbb{Z} \oplus \mathbb{Z}) \rtimes GL(2,\mathbb{Z})$. The method which we suggest to construct this isomorphism gives far more transparent picture of the structure of the automorphism group of the group ${\rm Heis}(3, \mathbb{Z}) $.
Zappa-Szép products of semigroups encompass both the self-similar group actions of Nekrashevych and the quasi-lattice-ordered groups of Nica. We use Li's construction of semigroup $C^*$-algebras to associate a $C^*$-algebra to Zappa-Szép products and give an explicit presentation of the algebra. We then define a quotient $C^*$-algebra that generalises the Cuntz-Pimsner algebras for self-similar actions. We indicate how known examples, previously viewed as distinct classes, fit into our unifying framework. We specifically discuss the Baumslag-Solitar groups, the binary adding machine, the semig…
The `colored Cube Dance' is an extension of Douthett's and Steinbach's Cube Dance graph, related to a monoid of binary relations defined on the set of major, minor, and augmented triads. This contribution explores the automorphism group of this monoid action, as a way to transform chord progressions. We show that this automorphism group is of order 7776 and is isomorphic to $({\mathbb{Z}_3}^4 \rtimes D_8) \rtimes (D_6 \times \mathbb{Z}_2)$. The size and complexity of this group makes it unwieldy: we therefore provide an interactive tool via a web interface based on common HTML/Javascript frame…
We prove a number of results on the survival of the type-I property under extensions of locally compact groups: (a) that given a closed normal embedding $\mathbb{N}\trianglelefteq\mathbb{E}$ of locally compact groups and a twisted action $(α,τ)$ thereof on a (post)liminal $C^*$-algebra $A$ the twisted crossed product $A\rtimes_{α,τ}\mathbb{E}$ is again (post)liminal and (b) a number of converses to the effect that under various conditions a normal, closed, cocompact subgroup $\mathbb{N}\trianglelefteq \mathbb{E}$ is type-I as soon as $\mathbb{E}$ is. This happens for instance if $\mathbb{N}$ i…
We establish results concerning the profinite completions of 3-manifold groups. In particular, we prove that the complement of the figure-eight knot $S^3-K$ is distinguished from all other compact 3-manifolds by the set of finite quotients of its fundamental group. In addition, we show that if $M$ is a compact 3-manifold with $b_1(M)=1$, and $π_1(M)$ has the same finite quotients as a free-by-cyclic group $F_r\rtimes\mathbb{Z}$, then $M$ has non-empty boundary, fibres over the circle with compact fibre, and $π_1(M)\cong F_r\rtimes_ψ\mathbb{Z}$ for some $ψ\in{\rm{Out}}(F_r)$.
Let $(X,T,μ)$ be a Cantor minimal sytem and $[[T]]$ the associated topological full group. We analyze $C^*_π([[T]])$, where $π$ is the Koopman representation attached to the action of $[[T]]$ on $(X,μ)$. Specifically, we show that $C^*_π([[T]])=C^*_π([[T]]')$ and that the kernel of the character $τ$ on $C^*_π([[T]])$ coming from weak containment of the trivial representation is a hereditary $C^*$-subalgebra of $C(X)\rtimes\mathbb{Z}$. Consequently, $\kerτ$ is stably isomorphic to $C(X)\rtimes\mathbb{Z}$, and $C^*_π([[T]]')$ is not AF. We also prove that if $G$ is a finitely generated, elementa…
Let $Γ$ be a discrete countable group. Consider the crossed product C$^\ast$-algebra $\mathfrak{R}(Γ) = C^{\ast}(Γ\rtimes l^{\infty}(Γ))$. Let $G$ be a larger discrete group, containing $Γ$ as an almost normal subgroup. Consequently $G$ acts by partial isomorphisms on $G$ and hence on $\mathfrak {R}(Γ)$. Let $\mathfrak{R}_G(Γ)$ be the crossed product $C^{\ast}$ - algebra $C^{\ast}(G \times (\mathfrak{R}(Γ))$. The C$^\ast$-algebra $\mathfrak{R}_G(Γ)$ has a natural representation into $\mathcal B(\ell ^2(Γ))$ and hence also admits a representation $Π_{\mathcal{Q}}$ into the Calkin algebra $\math…
In our previous paper, we introduced a hyperbolic jigsaw construction and constructed infinitely many non-commensurable, non-uniform, non-arithmetic lattices of $\mathrm{PSL}(2, \mathbb{R})$ with cusp set $\mathbb{Q} \cup \{\infty\}$ (called pseudomodular groups by Long and Reid), thus answering a question posed by Long and Reid. In this paper, we continue with our study of these jigsaw groups exploring questions of arithmeticity, pseudomodularity, and also related pseudo-euclidean and continued fraction algorithms arising from these groups. We also answer another question of Long and Reid by …
In \cite{BAMU}, an ergodic theorem à la Birkhoff-von Neumann for the action of the fundamental group of a compact negatively curved manifold on the boundary of its universal cover is proved. A quick corollary is the irreducibility of the associated unitary representation. These results are generalized \cite{BOYER} to the context of convex cocompact groups of isometries of a CAT(-1) space, using Theorem 4.1.1 of \cite{ROBLI}, with the hypothesis of non arithmeticity of the spectrum. We prove all the analog results in the case of the free group $\mathbb{F}_r$ of rank $r$ even if $\mathbb{F}_r$ i…
Let $C$ be a complex algebraic curve uniformised by a Fuchsian group $Γ$. In the first part of this paper we identify the automorphism group of the solenoid associated with $Γ$ with the Belyaev completion of its commensurator $\mathrm{Comm}(Γ)$ and we use this identification to show that the isomorphism class of this completion is an invariant of the natural Galois action of $\mathrm{Gal}(\mathbb C/\mathbb Q)$ on algebraic curves. In turn this fact yields a proof of the Galois invariance of the arithmeticity of $Γ$ independent of Kazhhdan's. In the second part we focus on the case in which $Γ$…
We study a subgroup $Fr(162\times 4)$ of SU(3) of order 648 which is an extension of $D(9,1,1;2,1,1)$ and whose generators arise from anyonic systems. We show that this group is isomorphic to a semi-direct product $(\mathbb{Z}/18\mathbb{Z}\times\mathbb{Z}/6\mathbb{Z})\rtimes S_3$ with respect to conjugation and we give a presentation of the group. We show that the group $D(18,1,1;2,1,1)$ from the series $(D)$ in the existing classification for finite SU(3)-subgroups is also isomorphic to a semi-direct product $(\mathbb{Z}/18\mathbb{Z}\times\mathbb{Z}/6\mathbb{Z})\rtimes S_3$, also with respect…
