Let B_n be the braid group on n strands, with n at least 4, and let Mod(S) be the extended mapping class group of the sphere with n+1 punctures. We show that the abstract commensurator of B_n is isomorphic to a semidirect product of Mod(S) with a group we refer to as the transvection subgroup, Tv(B_n). We also show that Tv(B_n) is itself isomorphic to a semidirect product of an infinite dimensional rational vector space with the multiplicative group of nonzero rational numbers.
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.
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.
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 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 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 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.
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 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.
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…
In this paper we consider the class of 2-dimensional Artin groups with connected, large type, triangle-free defining graphs (type CLTTF). We classify these groups up to isomorphism, and describe a generating set for the automorphism group of each such Artin group. In the case where the defining graph has no separating edge or vertex we show that the Artin group is not abstractly commensurable to any other CLTTF Artin group. If, moreover, the defining graph satisfies a further `vertex rigidity' condition, then the abstract commensurator group of the Artin group is isomorphic to its automorphism…
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.
Virtual braids are a combinatorial generalization of braids. We present abstract braids as equivalence classes of braid diagrams on a surface, joining two distinguished boundary components. They are identified up to isotopy, compatibility, stability and Reidemeister moves. We show that virtual braids are in a bijective correspondence with abstract braids. Finally we demonstrate that for any abstract braid, its representative of minimal genus is unique up to compatibility and Reidemeister moves. The genus of such a representative is thus an invariant for virtual braids. We also give a complete …
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.
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）
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 $\mathrm{Out}(W_n)$.
Let $N$ be a compact, connected, non-orientable surface of genus $ρ$ with $n$ boundary components, with $ρ\ge 5$ and $n \ge 0$, and let $\mathcal{M} (N)$ be the mapping class group of $N$. We show that, if $\mathcal{G}$ is a finite index subgroup of $\mathcal{M} (N)$ and $φ: \mathcal{G} \to \mathcal{M} (N)$ is an injective homomorphism, then there exists $f_0 \in \mathcal{M} (N)$ such that $φ(g) = f_0 g f_0^{-1}$ for all $g \in \mathcal{G}$. We deduce that the abstract commensurator of $\mathcal{M} (N)$ coincides with $\mathcal{M} (N)$.
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.
Let $\Out(F_n)$ denote the outer automorphism group of the free group $F_n$ with $n>3$. We prove that for any finite index subgroup $Γ<\Out(F_n)$, the group $\Aut(Γ)$ is isomorphic to the normalizer of $Γ$ in $\Out(F_n)$. We prove that $Γ$ is {\em co-Hopfian} : every injective homomorphism $Γ\to Γ$ is surjective. Finally, we prove that the abstract commensurator $\Comm(\Out(F_n))$ is isomorphic to $\Out(F_n)$.
We prove that if a right-angled Artin group $A_Γ$ is abstractly commensurable to a group splitting non-trivially as an amalgam or HNN-extension over $\mathbb{Z}^n$, then $A_Γ$ must itself split non-trivially over $\mathbb{Z}^k$ for some $k\le n$. Consequently, if two right-angled Artin groups $A_Γ$ and $A_Δ$ are commensurable and $Γ$ has no separating $k$-cliques for any $k\le n$ then neither does $Δ$, so "smallest size of separating clique" is a commensurability invariant. We also discuss some implications for issues of quasi-isometry. Using similar methods we also prove that for $n\ge 4$ the…
Let $Γ\leq \mathrm{Aut}(T_{d_1}) \times \mathrm{Aut}(T_{d_2})$ be a group acting freely and transitively on the product of two regular trees of degree $d_1$ and $d_2$. We develop an algorithm which computes the closure of the projection of $Γ$ on $\mathrm{Aut}(T_{d_t})$ under the hypothesis that $d_t \geq 6$ is even and that the local action of $Γ$ on $T_{d_t}$ contains $\mathrm{Alt}(d_t)$. We show that if $Γ$ is torsion-free and $d_1 = d_2 = 6$, exactly seven closed subgroups of $\mathrm{Aut}(T_6)$ arise in this way. We also construct two new infinite families of virtually simple lattices in …
