We prove that the abstract commensurator of a nonabelian free group, an infinite surface group, or more generally of a group that splits appropriately over a cyclic subgroup, is not finitely generated.
This applies in particular to all torsion-free word-hyperbolic groups with infinite outer automorphism group and abelianization of rank at least 2.
We also construct a finitely generated, torsion-free group which can be mapped onto Z and which has a finitely generated commensurator.
We show that the abstract commensurator of an S-arithmetic subgroup of a solvable algebraic group over Q is isomorphic to the Q-points of an algebraic group, and compare this with examples of nonlinear abstract commensurators of S-arithmetic groups in positive characteristic. In particular, we include a description of the abstract commensurator of the lamplighter group.
We prove that for any natural n>1, the abstract commensurator group of the Baumslag - Solitar group BS(1,n) is isomorphic to the group of 2 by 2 upper triangular matrices A over rational numbers with A_{11}=1. We also prove that for any finitely generated group G with the unique root property the natural homomorphisms Aut(G)--> Comm(G)--> QI(G) are embeddings.
P. Hall constructed a universal countable locally finite group U, determined up to isomorphism by two properties: every finite group C is a subgroup of U, and every embedding of C into U is conjugate in U. Every countable locally finite group is a subgroup of U. We prove that U is a subgroup of the abstract commensurator of a finite-rank nonabelian free group.
Any non-residually finite Baumslag-Solitar group has a non-residually finite image in the abstract commensuration of a nonabelian free group. This gives a new proof (avoiding Britton's Lemma) of the classification of residually finite Baumslag-Solitar groups.
In this paper we initiate a systematic study of the abstract commensurators of profinite groups. The abstract commensurator of a profinite group $G$ is a group $Comm(G)$ which depends only on the commensurability class of $G$. We study various properties of $Comm(G)$; in particular, we find two natural ways to turn it into a topological group. We also use $Comm(G)$ to study topological groups which contain $G$ as an open subgroup (all such groups are totally disconnected and locally compact). For instance, we construct a topologically simple group which contains the pro-2 completion of the Gri…
Let Gamma be a lattice in a simply-connected solvable Lie group. We construct a Q-defined algebraic group A such that the abstract commensurator of Gamma is isomorphic to A(Q) and Aut(Gamma) is commensurable with A(Z). Our proof uses the algebraic hull construction, due to Mostow, to define an algebraic group H so that commensurations of Gamma extend to Q-defined automorphisms of H. We prove an analogous result for lattices in connected linear Lie groups whose semisimple quotient satisfies superrigidity.
We prove that, aside from the obvious exceptions, the mapping class group of a compact orientable surface is not abstractly commensurable with any right-angled Artin group. Our argument applies to various subgroups of the mapping class group---the subgroups generated by powers of Dehn twists and the terms of the Johnson filtration---and additionally to the outer automorphism group of a free group and to certain linear groups.
The full solenoid over a topological space $X$ is the inverse limit of all finite covers. When $X$ is a compact Hausdorff space admitting a locally path connected universal cover, we relate the pointed homotopy equivalences of the full solenoid to the abstract commensurator of the fundamental group $π_1(X)$. The relationship is an isomorphism when $X$ is an aspherical CW complex. If $X$ is additionally a geodesic metric space and $π_1(X)$ is residually finite, we show that this topological model is compatible with the realization of the abstract commensurator as a subgroup of the quasi-isometr…
We show that if $π$ is the fundamental group of a 4-dimensional infrasolvmanifold then $-2\leq{def(π)}\leq0$, and give examples realizing each of these values. We also determine the abstract commensurators of such groups. Finally we show that if $G$ is a finitely generated group the kernel of the natural homomorphism from $G$ to its abstract commensurator $Comm(G)$ is locally nilpotent by locally finite, and is finite if $\mathrm{def}(G)>1$.
Let $Γ$ be the fundamental group of a surface of finite type and Comm$(Γ)$ be its abstract commensurator. Then Comm$(Γ)$ contains the solvable Baumslag--Solitar groups $\langle a ,b : a b a^{-1} = b^n \rangle$ for any $n > 1$. Moreover, the Baumslag--Solitar group $\langle a ,b : a b^2 a^{-1} = b^3 \rangle$ has an image in Comm$(Γ)$ that is not residually finite. Our proofs are computer-assisted. Our results also illustrate that finitely-generated subgroups of Comm$(Γ)$ are concrete objects amenable to computational methods. For example, we give a proof that $\langle a ,b : a b^2 a^{-1} = b^3 …
We study the structure of the commensurator of a virtually abelian subgroup $H$ in $G$, where $G$ acts properly on a $\mathrm{CAT}(0)$ space $X$. When $X$ is a Hadamard manifold and $H$ is semisimple, we show that the commensurator of $H$ coincides with the normalizer of a finite index subgroup of $H$. When $X$ is a $\mathrm{CAT}(0)$ cube complex or a thick Euclidean building and the action of $G$ is cellular, we show that the commensurator of $H$ is an ascending union of normalizers of finite index subgroups of $H$. We explore several special cases where the results can be strengthened and we…
We prove that if g and n are integers at least two, then the abstract commensurator of the braid group with n strands on a closed orientable surface of genus g is naturally isomorphic to the extended mapping class group of a compact orientable surface of genus g with n boundary components.
Let G be the automorphism group of a regular right-angled building X. The "standard uniform lattice" Γ_0 in G is a canonical graph product of finite groups, which acts discretely on X with quotient a chamber. We prove that the commensurator of Γ_0 is dense in G. This result was also obtained by Haglund. For our proof, we develop carefully a technique of "unfoldings" of complexes of groups. We use unfoldings to construct a sequence of uniform lattices Γ_n in G, each commensurable to Γ_0, and then apply the theory of group actions on complexes of groups to the sequence Γ_n. As further applicatio…
We prove that any ergodic measure-preserving action of an irreducible lattice in a semisimple group, with finite center and each simple factor having rank at least two, either has finite orbits or has finite stabilizers. The same dichotomy holds for many commensurators of such lattices. The above are derived from more general results on groups with the Howe-Moore property and property $(T)$. We prove similar results for commensurators in such groups and for irreducible lattices (and commensurators) in products of at least two such groups, at least one of which is totally disconnected.
On commensurators of free groups and free pro-p groups
Royal Holloway, University of London（伦敦大学皇家霍洛威学院）；University of Virginia（弗吉尼亚大学）；CNRS, UMPA - ENS Lyon（法国国家科学研究中心里昂高等师范学校）；Western Sydney University（西悉尼大学）；Università degli Studi di Firenze（佛罗伦萨大学）；Università degli Studi di Milano-Bicocca（米兰比可卡大学）
We describe the automorphism groups and the abstract commensurators of Houghton's groups. Then we give sharp estimates for the word metric of these groups and deduce that the commensurators embed into the corresponding quasi-isometry groups. As a further consequence, we obtain that the Houghton group on two rays is at least quadratically distorted in those with three or more rays.
A theorem of Farb and Handel asserts that for $N\ge 4$, the natural inclusion from $\mathrm{Out}(F_N)$ into its abstract commensurator is an isomorphism. We give a new proof of their result, which enables us to generalize it to the case where $N=3$. More generally, we give sufficient conditions on a subgroup $Γ$ of $\mathrm{Out}(F_N)$ ensuring that its abstract commensurator $\mathrm{Comm}(Γ)$ is isomorphic to its relative commensurator in $\mathrm{Out}(F_N)$. In particular, we prove that the abstract commensurator of the Torelli subgroup $\mathrm{IA}_N$ for all $N\ge 3$, or more generally any…
Suppose G is a non-free finitely generated Kleinian group without parabolics which is not a lattice and let C(G) denote the commensurator in PSL(2,C). We prove that if the limit set of G is not a round circle, then C(G) is discrete. Furthermore, G has finite index in C(G) unless G is a fiber group in which case C(G) is a lattice.
By examining commensurators of virtually cyclic groups, we show that for each natural number n, any locally finite-by-virtually cyclic group of cardinality aleph_n admits a finite dimensional classifying space with virtually cyclic stabilizers of dimension n+3. As a corollary, we prove that every elementary amenable group of finite Hirsch length and cardinality aleph_n admits a finite dimensional classifying space with virtually cyclic stabilizers.
We begin by showing that commensurators of Zariski dense subgroups of isometry groups of symmetric spaces of non-compact type are discrete provided that the limit set on the Furstenberg boundary is not invariant under the action of a (virtual) simple factor. In particular for rank one or simple Lie groups, Zariski dense subgroups with non-empty domain of discontinuity have discrete commensurators. This generalizes a Theorem of Greenberg for Kleinian groups. We then prove that for all finitely generated, Zariski dense, infinite covolume discrete subgroups of $Isom ({\mathbb{H}}^3)$, commensurat…
We say A is a quasi-normal subgroup of the group G if the commensurator of A in G is all of G. We develop geometric versions of commensurators in finitely generated groups. In particular, g is an element of the commensurator of A in G iff the Hausdorff distance between A and gA is finite. We show that a quasi-normal subgroup of a group is the kernel of a certain map, and a subgroup of a finitely generated group is quasi-normal iff the natural coset graph is locally finite. This last equivalence is particularly useful for deriving asymptotic results for finitely generated groups. Our primary go…
We determine the abstract commensurator com(F) of Thompson's group F and describe it in terms of piecewise linear homeomorphisms of the real line and in terms of tree pair diagrams. We show com (F) is not finitely generated and determine which subgroups of finite index in F are isomorphic to F. We show that the natural map from the commensurator group to the quasi-isometry group of F is injective.
We show that the commensurator of any quasiconvex abelian subgroup in a biautomatic group is small, in the sense that it has finite image in the abstract commensurator of the subgroup. Using this criterion we exhibit groups that are CAT(0) but not biautomatic. These groups also resolve a number of other questions concerning CAT(0) groups.
We prove an operator algebraic superrigidity statement for homomorphisms of irreducible lattices, and also their commensurators, in certain higher-rank groups into unitary groups of finite factors. This extends the authors' previous work regarding non-free measure-preserving actions, and also answers a question of Connes for such groups.
Let $\mathrm{Mod}(S)$ be the mapping class group of a compact connected orientable surface $S$, possibly with punctures and boundary components, with negative Euler characteristic. We prove that for any infinite virtually abelian subgroup $H$ of $\mathrm{Mod}(S)$, there is a subgroup $H'$ commensurable with $H$ such that the commensurator of $H$ equals the normalizer of $H'$. As a consequence we give, for each $n \geq 2$, an upper bound for the geometric dimension of $\mathrm{Mod}(S)$ for the family of abelian subgroups of rank bounded by $n$. These results generalize work by Juan-Pineda--Truj…
We study the lattice of finite-index extensions of a given finitely generated subgroup $H$ of a free group $F$. This lattice is finite and we give a combinatorial characterization of its greatest element, which is the commensurator of $H$. This characterization leads to a fast algorithm to compute the commensurator, which is based on a standard algorithm from automata theory. We also give a sub-exponential and super-polynomial upper bound for the number of finite-index extensions of $H$, and we give a language-theoretic characterization of the lattice of finite-index subgroups of $H$. Finally,…
We show that many normal subgroups of the braid group modulo its centre, and of the mapping class group of a sphere with marked points, have the property that their automorphism and abstract commensurator groups are mapping class groups of such spheres. As one application, we establish the automorphism groups of each term in the lower central series and derived series of the pure braid group. We also obtain new proofs of results of Dyer-Grossman and Orevkov, showing that the automorphism groups of the braid group and of its commutator subgroup are isomorphic. We then calculate the automorphism…
We show that the abstract commensurator of Thompson's group F is composed of four building blocks: two isomorphism types of simple groups, the multiplicative group of the positive rationals and a cyclic group of order two. The main result establishes the simplicity of a certain group of piecewise linear homeomorphisms of the real line.
We prove that a finitely generated pro-$p$ group acting on a pro-$p$ tree $T$ with procyclic edge stabilizers is the fundamental pro-$p$ group of a finite graph of pro-$p$ groups with edge and vertex groups being stabilizers of certain vertices and edges of $T$ respectively, in the following two situations: 1) the action is $n$-acylindrical, i.e., any non-identity element fixes not more than $n$ edges; 2) the group $G$ is generated by its vertex stabilizers. This theorem is applied to obtain several results about pro-$p$ groups from the class $\mathcal{L}$ defined and studied in [Math. Z. 267 …
This paper studies the rigidity properties of the abstract commensurator of the outer automorphism group of a universal Coxeter group of rank $n$, which is the free product $W_n$ of $n$ copies of $\mathbb{Z}/2\mathbb{Z}$. We prove that for $n\geq 5$ the natural map $\mathrm{Out}(W_n) \to \mathrm{Comm}(\mathrm{Out}(W_n))$ is an isomorphism and that every isomorphism between finite index subgroups of $\mathrm{Out}(W_n)$ is given by a conjugation by an element of $\mathrm{Out}(W_n)$.
Balanced groups and the virtually cyclic dimension of poly-surfaces groups
Centro de Ciencias Matemáticas, Universidad Nacional Autónoma de México（墨西哥国立自治大学数学科学中心）；Instituto de Matemáticas, Universidad Nacional Autónoma de México（墨西哥国立自治大学数学研究所）
We show that a nonempty family of $n$-generated subgroups of a pro-$p$ group has a maximal element. This suggests that 'Noetherian Induction' can be used to discover new features of finitely generated subgroups of pro-$p$ groups. To demonstrate this, we show that in various pro-$p$ groups $Γ$ (e.g. free pro-$p$ groups, nonsolvable Demushkin groups) the commensurator of a finitely generated subgroup $H \neq 1$ is the greatest subgroup of $Γ$ containing $H$ as an open subgroup. We also show that an ascending sequence of $n$-generated subgroups of a limit group must terminate (this extends the an…
Automorphism groups of locally finite trees provide a large class of examples of simple totally disconnected locally compact groups. It is desirable to understand the connections between the global and local structure of such a group. Topologically, the local structure is given by the commensurability class of a vertex stabiliser; on the other hand, the action on the tree suggests that the local structure should correspond to the local action of a stabiliser of a vertex on its neighbours. We study the interplay between these different aspects for the special class of groups satisfying Tits' in…
We study abstract group actions of locally compact Hausdorff groups on CAT(0) spaces. Under mild assumptions on the action we show that it is continuous or has a global fixed point. This mirrors results by Dudley and Morris-Nickolas for actions on trees. As a consequence we obtain a geometric proof for the fact that any abstract group homomorphism from a locally compact Hausdorff group into a torsion free CAT(0) group is continuous.
In this expanded account of a talk given at the Oberwolfach Arbeitsgemeinschaft "Totally Disconnected Groups", October 2014, we discuss results of Nikolay Nikolov and Dan Segal on abstract quotients of compact Hausdorff topological groups, paying special attention to the class of finitely generated profinite groups. Our primary source is a paper, entitled "Generators and commutators in finite groups; abstract quotients of compact groups", that was published by Nikolov and Segal in 2012. Sidestepping all difficult and technical proofs, we present a selection of accessible arguments to illuminat…
We show that every abstract homomorphism $φ$ from a locally compact group $L$ to a graph product $G_Γ$, endowed with the discrete topology, is either continuous or $φ(L)$ lies in a 'small' parabolic subgroup. In particular, every locally compact group topology on a graph product whose graph is not 'small' is discrete. This extends earlier work by Morris-Nickolas. We also show the following. If $L$ is a locally compact group and if $G$ is a discrete group which contains no infinite torsion group and no infinitely generated abelian group, then every abstract homomorphism $φ:L\to G$ is either con…
Consider the Iwasawa decomposition of the real semisimple Lie group. The purpose of this paper is to define the Fourier transform in order to obtain the Plancherel theorem on its maxima solvable Lie group. Besides, we prove the existence theorems for the invariant differential operators on this solvable. To this end, we give a classification of all left ideals of its group algebra
For any surface $Σ$ of infinite topological type, we study the Torelli subgroup ${\mathcal I}(Σ)$ of the mapping class group ${\rm MCG}(Σ)$, whose elements are those mapping classes that act trivially on the homology of $Σ$. Our first result asserts that ${\mathcal I}(Σ)$ is topologically generated by the subgroup of ${\rm MCG}(Σ)$ consisting of those elements in the Torelli group which have compact support. In particular, using results of Birman, Powell, and Putman we deduce that ${\mathcal I}(Σ)$ is topologically generated by separating twists and bounding pair maps. Next, we prove the abstr…
Let N be a normal subgroup of a finite group G. We prove that under certain (unavoidable) conditions the subgroup [N,G] is a product of commutators [N,y] (with prescribed values of y from a given set Y) of length bounded by a function of d(G) and |Y| only. This has several applications: 1. A new proof that G^n is closed (and hence open) in any finitely generated profinite group G. 2. A finitely generated abstract quotient of a compact Hausdorff group must be finite. 3. Let G be a topologically finitely generated compact Hausdorff group. Then G has a countably infinite abstract quotient if and …
