arXiv research

New lattices in higher dimensions have dense surface subgroups.

The paper examines subgroup separability for surface and virtual braid groups.

We prove that every finitely generated Kleinian group that contains a finite, non-cyclic subgroup either is finite or virtually free or contains a surface subgroup. Hence, every arithmetic Kleinian group contains a surface subgroup.

The study finds surface subgroups in cocompact lattices of $H^{2n}$ for $n\geq2$.

Minimal involutions generate a subgroup of nonorientable surfaces.

The paper controls the geometry of surface subgroups in specific Kleinian groups.

The study finds that certain hyperbolic manifolds contain subgroups isomorphic to surface groups.

New combinatorial structures represent subgroups of surface groups, analogous to Stallings core graphs.

New proof and description of commutator subgroups for free and surface groups.

We give several sufficient conditions for a double of a free group along a cyclic subgroup to contain a surface subgroup.

We show that every finitely-generated free subgroup of a right-angled, co-compact Kleinian reflection group is contained in a surface subgroup.

A random group contains many quasiconvex surface subgroups.

The study bounds the number of quasi-Fuchsian surface subgroups in hyperbolic 3-manifolds.

The study restricts normal subgroups of Kähler groups, proving specific cases and general restrictions.

The main result of this article is a refinement of the well-known subgroup separability results of Hall and Scott for free and surface groups. We show that for any finitely generated subgroup, there is a finite dimensional representation of the free or surface group that separates the subgroup in the induced Zariski to…

Study shows conjugacy of torsion in genus 2 surfaces.

A random graph of free groups contains a surface subgroup

Study on infinite-type surfaces shows stable commutator length is continuous and defines open subgroups.

Researchers determine the rational abelianization of a subgroup of mapping class groups.

The paper classifies fixed subgroups of endomorphisms in free-abelian times surface groups.

Pseudo-Anosov subgroups in surface bundles over tori are convex cocompact.

Classifies hyperbolic groups with surface-like boundaries.

Classifies surface Houghton groups and their subgroups up to certain equivalences.

We consider the question of which right-angled Artin groups contain closed hyperbolic surface subgroups. It is known that a right-angled Artin group $A(K)$ has such a subgroup if its defining graph $K$ contains an $n$-hole (i.e. an induced cycle of length $n$) with $n\geq 5$. We construct another eight "forbidden" grap…

Study shows hyperbolic subgroups can be free products of surface and free groups.

Odd-dimensional SL(n,Q) contains dense surface subgroups.

The study restricts Anosov subgroups of Sp(2n,R) based on subset Θ.

Birman-Lubotzky-McCarthy proved that any abelian subgroup of the mapping class groups for orientable surfaces is finitely generated. We apply Birman-Lubotzky-McCarthy's arguments to the mapping class groups for non-orientable surfaces. We especially find a finitely generated group isomorphic to a given torsion-free sub…

We prove that various subgroups of the mapping class group $Mod(Σ)$ of a surface $Σ$ are at least exponentially distorted. Examples include the Torelli group (answering a question of Hamenstadt), the "point-pushing" and surface braid subgroups, and the Lagrangian subgroup. Our techniques include a method to compute low…

We determine the largest (i.e. smallest index) characteristic subgroup of surface groups not containing any simple loops.

Let Gamma < PSL_2(C) be discrete, cofinite volume, and noncocompact. We prove that for all K > 1, there is a subgroup H < Gamma that is K-quasiconformally conjugate to a discrete cocompact subgroup of PSL_2(R). Along with previous work of Kahn and Markovic, this proves that every finite covolume Kleinian group has a ne…

In the vein of Bonfert-Taylor, Bridgeman, Canary, and Taylor we introduce the notion of quasiconformal homogeneity for closed oriented hyperbolic surfaces restricted to subgroups of the mapping class group. We find uniform lower bounds for the associated quasiconformal homogeneity constants across all closed hyperbolic…

There is an established bijection between finite-index subgroups Gamma of Gamma(2) and bipartite graphs on surfaces, or, equivalently, certain triples of permutations. We utilize this relationship to study both congruence and noncongruence subgroups in terms of the corresponding graphs. We show some elementary criteria…

We study smooth complex hypersurfaces in direct products of closed hyperbolic Riemann surfaces and give a classification in terms of their fundamental groups. This answers a question of Delzant and Gromov on subvarieties of products of Riemann surfaces in the smooth codimension one case. We also answer Delzant and Grom…

Proves congruence subgroup property for mapping class groups of hyperbolic surfaces.

New subgroups of mapping class groups constructed for infinite-type surfaces.

Let G be a word-hyperbolic group, obtained as a graph of free groups amalgamated along cyclic subgroups. If H_2(G;Q) is nonzero, then G contains a closed hyperbolic surface subgroup. Moreover, the unit ball of the Gromov-Thurston norm on H_2(G;R) is a finite-sided rational polyhedron.

We prove that the handlebody subgroup of the Torelli group of an orientable surface is generated by genus one BP-maps. As an application, we give a normal generating set for the handlebody subgroup of the level $d$ mapping class group of an orientable surface.

If F is a surface with boundary, then a finitely generated subgroup without peripheral elements of G = π_1(F) can be separated from finitely many other elements of G by a finite index subgroup of G corresponding to a finite cover F' with the same number of boundary components as F .

This paper provides an infinite presentation for a subgroup of mapping class groups of non-orientable surfaces.

Study counts surface subgroups in curved 3D manifolds.

Researchers found that the twist subgroup can be generated by two elements for certain surface genera.

Characterizes groups arising as fixed subgroups of RAAG automorphisms.

The study examines conditions for symmetric and alternating subgroups in mapping class groups of surfaces.

We propose a sliding surface for systems on the Lie group $SO(3)\times \mathbb{R}^3$ . The sliding surface is shown to be a Lie subgroup. The reduced-order dynamics along the sliding subgroup have an almost globally asymptotically stable equilibrium. The sliding surface is used to design a sliding-mode controller for t…

Study Farrell cohomology for non-orientable surfaces, classifying subgroup conjugacy.

Explicitly bounds the spectral gap for Schottky subgroups of SL(2,Z).

The study classifies normal subgroups of mapping class groups of surfaces with Cantor subsets.