We show that the outer automorphism group of a polycyclic-by-finite group is an arithmetic group. This result follows from a detailed structural analysis of the automorphism groups of such groups. We use an extended version of the theory of the algebraic hull functor initiated by Mostow. We thus make applicable refined methods from the theory of algebraic and arithmetic groups. We also construct examples of polycyclic-by-finite groups which have an automorphism group which does not contain an arithmetic group of finite index. Finally we discuss applications of our results to the groups of homotopy self-equivalences of K(Γ, 1)-spaces and obtain an extension of arithmeticity results of Sullivan in rational homotopy theory.
Let $G$ be a finite solvable group, given through a refined consistent polycyclic presentation, and $α$ an automorphism of $G$, given through its images of the generators of $G$. In this paper, we discuss algorithms for computing the order of $α$ as well as the cycle length of a given element of $G$ under $α$. We give correctness proofs and discuss the theoretical complexity of these algorithms. Along the way, we carry out detailed complexity analyses of several classical algorithms on finite polycyclic groups.
In this paper, we show that any subgroup of a semidirect product of Z^n with Z has bounded packing as long as the action of Z on Z^n is by diagonalizable automorphisms all of whose eigenvalues are real. We use this result to show that any subgroup in a polycyclic group of length 3 or less has bounded packing. We also introduce the notion of coset growth and obtain a bound for the coset growth of subgroup H=<t> in the semidirect product of Z^2 with Z.
Let F be a local field of positive characteristic, and let G be either a Heisenberg group over F, or a certain (nonabelian) two-dimensional unipotent group over F. If H is an arithmetic subgroup of G, we provide an explicit description of every automorphism of H. From this description, it follows that every automorphism of H virtually extends to a virtual automorphism of G.
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…
We show that for every g > 1 there is a compact arithmetic Riemann surface of genus g with at least 4(g-1) automorphisms, and that this lower bound is attained by infinitely many genera, the smallest being 24.
Let $R(ϕ)$ be the number of $ϕ$-conjugacy (or Reidemeister) classes of an endomorphism $ϕ$ of a group $G$. We prove for several classes of groups (including polycyclic) that the number $R(ϕ)$ is equal to the number of fixed points of the induced map of an appropriate subspace of the unitary dual space $\widehat G$, when $R(ϕ)<\infty$. Applying the result to iterations of $ϕ$ we obtain Gauss congruences for Reidemeister numbers. In contrast with the case of automorphisms, studied previously, we have a plenty of examples having the above finiteness condition, even among groups with $R_\infty$ pr…
It was previously shown by Grunewald and Lubotzky that the automorphism group of a free group, $\text{Aut}(F_n)$, has a large collection of virtual arithmetic quotients. Analogous results were proved for the mapping class group by Looijenga and by Grunewald, Larsen, Lubotzky, and Malestein. In this paper, we prove analogous results for the automorphism group of a right-angled Artin group for a large collection of defining graphs. As a corollary of our methods we produce new virtual arithmetic quotients of $\text{Aut}(F_n)$ for $n \geq 4$ where $k$th powers of all transvections act trivially fo…
We show that Out(G) is residually finite if G is a one-ended group that is hyperbolic relative to virtually polycyclic subgroups. More generally, if G is one-ended and hyperbolic relative to proper residually finite subgroups, the group of outer automorphisms preserving the peripheral structure is residually finite. We also show that Out(G) is virtually p-residually finite for every prime p if G is one-ended and toral relatively hyperbolic, or infinitely-ended and virtually p-residually finite.
Let $G=G_1\ast\dots\ast G_k\ast F$ be a countable group which splits as a free product, where all groups $G_i$ are freely indecomposable and not isomorphic to $\mathbb{Z}$, and $F$ is a finitely generated free group. If for all $i\in\{1,\dots,k\}$, both $G_i$ and its outer automorphism group $\text{Out}(G_i)$ satisfy the Tits alternative, then $\text{Out}(G)$ satisfies the Tits alternative. As an application, we prove that the Tits alternative holds for outer automorphism groups of right-angled Artin groups, and of torsion-free groups that are hyperbolic relative to a finite family of virtuall…
Automorphisms of relatively hyperbolic groups and the Farrell--Jones Conjecture
Laboratoire de Mathématiques d’Orsay, Université Paris-Saclay（巴黎萨克雷大学）；Université de Caen Normandie, CNRS, Laboratoire de Mathématiques Nicolas Oresme（诺曼底大学）；Rheinische Friedrich-Wilhelms-Universität Bonn（波恩大学）；University of Oxford（牛津大学）；Univ. Lyon, ENS de Lyon, UMPA UMR 5669（里昂大学）
The goal of this paper is to construct and describe certain arithmetic subgroups of the automorphism group of a partially commutative group. More precisely, given an arbitrary finite graph $Γ$ we construct an arithmetic subgroup $St(L(G))$, represented as a subgroup of $GL(n,Z)$, where $n$ is the number of vertices of the graph $Γ$. In the last section of the paper we give a description of the decomposition of the group of automorphisms $St^{conj}(L(G))$ as a semidirect product of the group of conjugating automorphisms $Conj(G)$ and $St(L(G))$. This result is closely related to Theorem 1.4 of …
The automorphism tower of a group is obtained by computing its automorphism group, the automorphism group of THAT group, and so on, iterating transfinitely. Each group maps canonically into the next using inner automorphisms, and so at limit stages one can take a direct limit and continue the iteration. The tower is said to terminate if a fixed point is reached, that is, if a group is reached which is isomorphic to its automorphism group by the natural map. This occurs if a complete group is reached, one which is centerless and has only inner automorphisms. Wielandt [1939] proved the classical…
In this paper we study a certain class of polycyclic groups. We outline a method for constructing a poly-$\mathbb{Z}$ group $G_n$ by describing a process for selecting maps that are used to extend $G_i$ to $G_{i+1}$ for $1 \leq i \leq n-1$ and describe the multiplicative structure and automorphism groups of some poly-$\mathbb{Z}$ groups up to $G_3$.
We prove structure theorems for the moduli stack of elliptic curves equipped with $G$-structures, where $G$ is a finite 2-generated metabelian group. In particular, we show that if $G$ has exponent $e$, then there is a subgroup $H\le GL_2(\mathbb{Z}/e)$ such that $G$-structures on elliptic curves $E$ are equivalent to "congruence structures of level $H$". Our methods are almost entirely group theoretic. Let $\widehat{M}$ denote the free profinite metabelian group of rank 2, then along the way we prove a decomposition of $Out(\widehat{M})$ as an internal semi-direct product of the subgroup of "…
Let $F_n$ be the free group on $n\ge 2$ elements and $\A(F_n)$ its group of automorphisms. In this paper we present a rich collection of linear representations of $\A(F_n)$ arising through the action of finite index subgroups of it on relation modules of finite quotient groups of $F_n$. We show (under certain conditions) that the images of our representations are arithmetic groups.
We give simple examples of Kazhdan groups with infinite outer automorphism groups. This answers a question of Paulin, independently answered by Ollivier and Wise by completely different methods. As arithmetic lattices in (non-semisimple) Lie groups, our examples are in addition finitely presented. We also use results of Abels about compact presentability of p-adic groups to exhibit a finitely presented non-Hopfian Kazhdan group. This answers a question of Ollivier and Wise.
We give a cohomological criterion for existence of outer automorphisms of a semisimple algebraic group over an arbitrary field. This criterion is then applied to the special case of groups of type D_2n over a global field, which completes some of the main results from the paper "Weakly commensurable arithmetic groups and isospectral locally symmetric spaces" (Pub. Math. IHES, 2009) by Prasad and Rapinchuk and gives a new proof of a result from another paper by the same authors.
We prove a rigidity theorem for semi-arithmetic Fuchsian groups: If $Γ_1$, $Γ_2$ are two semi-arithmetic lattices in $\mathrm{PSL}(2,\mathbb{R})$ virtually admitting modular embeddings and $f\colonΓ_1\toΓ_2$ is a group isomorphism that respects the notion of congruence subgroups, then $f$ is induced by an inner automorphism of $\mathrm{PGL}(2,\mathbb{R})$.
The automorphism groups of integral Lorentzian lattices act by isometries on hyperbolic space with finite covolume. In the case of reflective integral lattices, the automorphism groups are commensurable to arithmetic hyperbolic reflection groups. However, for a fixed dimension, there is only finitely many reflective integral Lorentzian lattices, and these can only occur in small dimensions. The goal of this note is to construct embeddings of low-dimensional integral Lorentzian lattices into unimodular Lorentzian lattices associated to right-angled reflection groups. As an application, we const…
For a wide class of groups including polycyclic and finitely generated polynomial growth groups it is proved that the Reidemeister number of an automorphism f is equal to the number of finite-dimensional fixed points of the induced map f^ on the unitary dual, if one of these numbers is finite. This theorem is a natural generalization of the classical Burnside-Frobenius theorem to infinite groups. This theorem also has important consequences in topological dynamics and in some sense is a reply to a remark of J.-P. Serre. The main technical results proved in the paper yield a tool for a further …
Given a free group $F_k$ of rank $k\ge 2$ with a fixed set of free generators we associate to any homomorphism $ϕ$ from $F_k$ to a group $G$ with a left-invariant semi-norm a generic stretching factor, $λ(ϕ)$, which is a non-commutative generalization of the translation number. We concentrate on the situation when $ϕ:F_k\to Aut(X)$ corresponds to a free action of $F_k$ on a simplicial tree $X$, in particular, when $ϕ$ corresponds to the action of $F_k$ on its Cayley graph via an automorphism of $F_k$. In this case we are able to obtain some detailed ``arithmetic'' information about the possibl…
In the first three sections, we develop some basic facts about hypergeometric sheaves on the multiplicative group ${\mathbb G}_m$ in characteristic $p >0$. In the fourth and fifth sections, we specialize to quite special classses of hypergeomtric sheaves. We give relatively "simple" formulas for their trace functions, and a criterion for them to have finite monodromy. In the next section, we prove that three of them have finite monodromy groups.We then give some results on finite complex linear groups. We next use these group theoretic results to show that one of our local systems, of rank $24…
This is a survey of the Kawamata-Morrison cone conjecture on the structure of Calabi-Yau varieties and more generally Calabi-Yau pairs. We discuss the proof of the cone conjecture for algebraic surfaces, with plenty of examples. We show that the automorphism group of a K3 surface need not be commensurable with an arithmetic group, which answers a question by Mazur.
We solve Dehn's isomorphism problem for virtually torsion-free relatively hyperbolic groups with nilpotent parabolic subgroups. We do so by reducing the isomorphism problem to three algorithmic problems in the parabolic subgroups, namely the isomorphism problem, separation of torsion (in their outer automorphism groups) by congruences, and the mixed Whitehead problem, an automorphism group orbit problem. The first step of the reduction is to compute canonical JSJ decompositions. Dehn fillings and the given solutions of the algorithmic problems in the parabolic groups are then used to decide if…
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 internal Lie algebras in the category of subshifts on a fixed group -- or Lie algebraic subshifts for short. We show that if the acting group is virtually polycyclic and the underlying vector space has dense homoclinic points, such subshifts can be recoded to have a cellwise Lie bracket. On the other hand there exist Lie algebraic subshifts (on any finitely-generated non-torsion group) with cellwise vector space operations whose bracket cannot be recoded to be cellwise. We also show that one-dimensional full vector shifts with cellwise vector space operations can support infinitely ma…
A hyperbolic lattice is called \textit{$(1{,}2)$-reflective} if its automorphism group is generated by $1$- and $2$-reflections up to finite index. In this paper we prove that the fundamental polyhedron of a $\mathbb{Q}$-arithmetic cocompact reflection group in the three-dimensional Lobachevsky space contains an edge such that the distance between its framing faces is small enough. Using this fact we obtain a classification of $(1{,}2)$-reflective anisotropic hyperbolic lattices of rank $4$.
