Commensurable Groups Research Articles

Conjugacy growth in the higher Heisenberg groups

AbstractWe calculate asymptotic estimates for the conjugacy growth function of finitely generated class 2 nilpotent groups whose derived subgroups are infinite cyclic, including the so-called higher Heisenberg groups. We prove that these asymptotics are stable when passing to commensurable groups, by understanding their twisted conjugacy growth. We also use these estimates to prove that, in certain cases, the conjugacy growth series cannot be a holonomic function.

Arithmetic Subgroups and Applications

Arithmetic subgroups are an important source of discrete groups acting freely on manifolds. We need to know that there exist many torsion-free 푺푺L(ퟐퟐ,ℝ) is an “arithmetic” subgroup of 푺푺L(ퟐퟐ,ℝ). The other arithmetic subgroups are not as obvious, but they can be constructed by using quaternion algebras. Replacing the quaternion algebras with larger division algebras yields many arithmetic subgroups of 푺푺L(풏풏,ℝ), with 풏풏>2. In fact, a calculation of group cohomology shows that the only other way to construct arithmetic subgroups of 푺푺L(풏풏, ℝ) is by using arithmetic groups. In this paper justifies Commensurable groups, and some definitions and examples,ℝ-forms of classical simple groups over ℂ, calculating the complexification of each classical group, Applications to manifolds. Let us start with 푺푺푺푺(푛푛,ℂ). This is already a complex Lie group, but we can think of it as a real Lie group of twice the dimension. As such, it has a complexification.

Big Torelli groups: generation and commensuration

For any surface \Sigma of infinite topological type, we study the Torelli subgroup \mathcal I(\Sigma) of the mapping class group MCG (\Sigma) , whose elements are those mapping classes that act trivially on the homology of \Sigma . Our first result asserts that {\mathcal I}(\Sigma) is topologically generated by the subgroup of MCG (\Sigma) consisting of those elements in the Torelli group which have compact support. Next, we prove the abstract commensurator group of {\mathcal I}(\Sigma) coincides with MCG (\Sigma) . This extends the results for finite-type surfaces [9, 6, 7, 16] to the setting of infinite-type surfaces.

Normal subgroups of mapping class groups and the metaconjecture of Ivanov

We prove that if a normal subgroup of the extended mapping class group of a closed surface has an element of sufficiently small support, then its automorphism group and abstract commensurator group are both isomorphic to the extended mapping class group. The proof relies on another theorem we prove, which states that many simplicial complexes associated to a closed surface have automorphism group isomorphic to the extended mapping class group. These results resolve the metaconjecture of N. V. Ivanov, which asserts that any “sufficiently rich” object associated to a surface has automorphism group isomorphic to the extended mapping class group, for a broad class of such objects. As applications, we show: (1) right-angled Artin groups and surface groups cannot be isomorphic to normal subgroups of mapping class groups containing elements of small support, (2) normal subgroups of distinct mapping class groups cannot be isomorphic if they both have elements of small support, and (3) distinct normal subgroups of the mapping class group with elements of small support are not isomorphic. Our results also suggest a new framework for the classification of normal subgroups of the mapping class group.

God, King, and Subject: On the Development of Composite Political Cultures in the Western Himalaya, circa 1800–1900

The history of British rule in the Indian Himalaya exemplifies the mutual enforcement of social identities and political cultures in modern South Asia. For the Khas ethnic majority of the Himachal Pradesh–Uttarakhand borderland, the colonial power's differentiation between “secular” and “religious” authorities engendered the division of substantially commensurable groups into “caste Hindu” and “tribal” societies. In demarcating borders along the “natural barrier” between the states, the British had severed a politically potent grassroots theocracy from its underlings, consolidated the fragmentation of the Shimla Hill States, and ultimately encouraged the development of a composite political cultures that complemented Khas traditions with Brahmanical creeds from the plains.

On universal minimal proximal flows of topological groups

In this paper, we show that the action of a characteristically simple, non-extremely amenable (non-strongly amenable, non-amenable) group on its universal minimal (minimal proximal, minimal strongly proximal) flow is effective. We present necessary and sufficient conditions, for the action of a topological group with trivial center on its universal minimal proximal flow, to be effective. A theorem of Furstenberg about the isomorphism of the universal minimal proximal flows of a discrete group and its subgroups of finite index ([Theorem~II.4.4]) is strengthened. Finally, for a pair of groups $H < G$ the same method is applied in order to extend the action of $H$ on its universal minimal proximal flow to an action of its commensurator group $\mathrm{Comm}_G(H)$.

Addendum to ‘‘On integral quadratic forms having commensurable groups of automorphisms’’

Addendum to ‘‘On integral quadratic forms having commensurable groups of automorphisms’’

On integral quadratic forms having commensurable groups of automorphisms

We introduce two notions of equivalence for rational quadratic forms. Two $n$-ary rational quadratic forms are commensurable if they possess commensurable groups of automorphisms up to isometry. Two $n$-ary rational quadratic forms $F$ and $G$ are projectivelly equivalent if there are nonzero rational numbers $r$ and $s$ such that $rF$ and $sG$ are rationally equivalent. It is shown that if $F$\ and $G$\ have Sylvester signature $\{-,+,+,...,+\}$ then $F$\ and $G$\ are commensurable if and only if they are projectivelly equivalent. The main objective of this paper is to obtain a complete system of (computable) numerical invariants of rational $n$-ary quadratic forms up to projective equivalence. These invariants are a variation of Conway's $p$-excesses. Here the cases $n$ odd and $n$ even are surprisingly different. The paper ends with some examples

Abstract Commensurators of Solvable Baumslag–Solitar Groups

We prove that for any n ≥ 2, the abstract commensurator group of the Baumslag–Solitar group BS(1, n) is isomorphic to the subgroup of GL2(ℚ). 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.

Simple locally compact groups acting on trees and their germs of automorphisms

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' independence property. We show that such a group has few open subgroups if and only if it acts locally primitively. Moreover, we show that it always admits many germs of automorphisms which do not extend to automorphisms, from which we deduce a negative answer to a question by George Willis. Finally, under suitable assumptions, we compute the full group of germs of automorphisms; in some specific cases, these turn out to be simple and compactly generated, thereby providing a new infinite family of examples which generalise Neretin's group of spheromorphisms. Our methods describe more generally the abstract commensurator group for a large family of self-replicating profinite branch groups.

Commensurations and subgroups of finite index of Thompson’s groupF

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.

Automorphisms and abstract commensurators of 2–dimensional Artin groups

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 group and generated by inner automorphisms, graph automorphisms (induced from automorphisms of the defining graph), and the involution which maps each standard generator to its inverse. We observe that the techniques used here to study automorphisms carry over easily to the Coxeter group situation. We thus obtain a classification of the CLTTF type Coxeter groups up to isomorphism and a description of their automorphism groups analogous to that given for the Artin groups.

The geometry of surface-by-free groups

We show that every word hyperbolic, surface-by-(noncyclic) free group \( \Gamma \) is as rigid as possible: the quasi-isometry group of \( \Gamma \) equals the abstract commensurator group Comm(\( \Gamma \)), which in turn contains \( \Gamma \) as a finite-index subgroup. As a corollary, two such groups are quasiisometric if and only if they are commensurable, and any finitely-generated group quasi-isometric to \( \Gamma \) must be weakly commensurable with \( \Gamma \). We use quasi-isometries to compute Comm(\( \Gamma \)) explicitly, an example of how quasi-isometries can actually detect finite-index information. The proofs of these theorems involve ideas from coarse topology, Teichmuller geometry, pseudo-Anosov dynamics, and singular SOLV geometry.