arXiv research

Abstract commensurators of ${ m{Aut}}(F_N)$ and ${ m{IA}}_N$ are the same.

Abstract commensurators linked to topological models of solenoids.

Let $\mathcal{X}_S$ denote the class of spaces homeomorphic to two closed orientable surfaces of genus greater than one identified to each other along an essential simple closed curve in each surface. Let $\mathcal{C}_S$ denote the set of fundamental groups of spaces in $\mathcal{X}_S$. In this paper, we characterize t…

Abstract commensurators of surface groups contain specific Baumslag-Solitar groups and are computationally accessible.

New proof confirms subgroup commensurators for $N\geq 3$.

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 f…

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 tha…

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.

The paper shows that for Coxeter groups, the commensurator of outer automorphisms is rigid.

Conditions for commensurability in specific groups are derived.

Study shows topological rigidity fails for certain Coxeter group quotients.

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 Johnso…

The paper explores normal subgroups of braid groups and mapping class groups.

Injective homomorphisms of non-orientable mapping class groups are conjugate.

Artin groups of types $F_4$ and $H_4$ are not commensurable with $D_4$.

Study of Torelli groups on infinite-type surfaces, focusing on generation and commensuration.

Study shows some surface group amalgams can't act on 3-manifolds.

We compute the automorphism groups of the Torelli complex and the complex of separating curves for all but finitely many compact orientable surfaces. As an application, we show that the abstract commensurators of the Torelli group and the Johnson kernel for such surfaces are naturally isomorphic to the extended mapping…

The study examines free products of hyperbolic manifold groups and their model geometries.

Many normal subgroups of mapping class groups are geometric.

Classifies groups quasi-isometric to torsion-generated ones in specific hyperbolic groups.

Quadratic differentials grouped by commensurability, each with a unique representative.

We investigate commensurability classes of hyperbolic knot complements in the generic case of knots without hidden symmetries. We show that such knot complements which are commensurable are cyclically commensurable, and that there are at most $3$ hyperbolic knot complements in a cyclic commensurability class. Moreover …

Defines fibered commensurability for outer automorphisms of free groups, proving uniqueness of minimal elements.

This paper classifies commensurability of Deligne-Mostow lattices.

For each closed orientable surface we introduce a simplical complex with some additional structure which is a version of the complex of curves of this surface adjusted to investigation of its Torelli group. We call this complex the Torelli geometry of our surface and prove that every automorphism of the Torelli geometr…

Random quotients of mapping class groups have rigid properties.

We show that a hyperbolic 2-bridge knot complement is the unique knot complement in its commensurability class. We also discuss constructions of commensurable hyperbolic knot complements and put forth a conjecture on the number of hyperbolic knot complements in a commensurability class.

Artin groups of spherical type are commensurable if they have the same irreducible components and rank.

This paper initiates a systematic study of the relation of commensurability of surface automorphisms, or equivalently, fibered commensurability of 3-manifolds fibering over the circle. We show that every hyperbolic fibered commensurability class contains a unique minimal element, whereas the class of Seifert manifolds …

Topology on commensurability classes of hyperbolic 3-manifolds studied.

New complex hyperbolic lattices discovered from triangle groups.

Torelli subgroup rigidity in Out(F_N) proven for N≥4.

Study shows counterexamples to group boundary and commensurability questions.

The paper shows commensurators of certain subgroups are discrete.

The paper explores conditions for topological rigidity in quotients of the Davis complex.

Study on achiral Sol 3-manifolds with density results.

Proves normal subgroups of mapping class groups have specific properties.

The study examines how non-commensurable surfaces can share length spectra.

Study mutant pairs of hyperbolic polyhedra, focusing on commensurability.

The paper explores cusp types in hyperbolic 4-manifolds and their commensurability classes.

Gopal Prasad and A. S. Rapinchuk defined a notion of weakly commensurable lattices in a semisimple group, and gave a classification of weakly commensurable Zariski dense subgroups. A motivation was to classify pairs of locally symmetric spaces isospectral with respect to the Laplacian on functions. For this, in higher …

Discrete commensurators of certain subgroups of PSL2(R) proven.

This paper describes a general algorithm for finding the commensurator of a non-arithmetic cusped hyperbolic manifold, and for deciding when two such manifolds are commensurable. The method is based on some elementary observations regarding horosphere packings and canonical cell decompositions. For example, we use this…

We parametrize the commensurability classes of curves on Shimura surfaces that are totally geodesic, i.e., the commensurability classes of so-called $\mathbb{C}$-Fuchsian subgroups. In particular, if a Shimura surface contains one commensurability class of totally geodesic curves, it contains infinitely many.

The study finds infinite commensurability classes of hyperbolic manifolds with Salem number lengths.

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 dens…

Two flows are topologically almost commensurable if, up to removing finitely many periodic orbits and taking finite coverings, they are topologically equivalent. We prove that all suspensions of automorphisms of the 2-dimensional torus and all geodesic flows on unit tangent bundles to hyperbolic 2-orbifolds are pairwis…