arXiv research

Proves Congruence Subgroup Property for two types of groups.

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

The study examines groups with a specific automorphism property using BNS-invariant.

We show that any subgroup of a (virtually) nilpotent-by-polycyclic group satisfies the bounded packing property of Hruska-Wise. In particular, the same is true about metabelian groups and linear solvable groups. However, we find an example of a finitely generated solvable group of derived length 3 which admits a finite…

In this paper we create many examples of hyperbolic groups with subgroups satisfying interesting finiteness properties. We give the first examples of subgroups of hyperbolic groups which are of type $FP_2$ but not finitely presented. We give uncountably many groups of type $FP_2$ with similar properties to those subgro…

There is an infinitely generated free subgroup of the smooth knot concordance group with the property that no nontrivial element in this subgroup can be represented by an alternating knot. This subgroup has the further property that every element is represented by a topologically slice knot.

Study complex hyperbolic lattices and their subgroups, proving new finiteness properties.

Study shows conjugacy of torsion in genus 2 surfaces.

Example found of subgroup not a lattice in product of Lie groups

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

Positive Thompson links are proven for oriented subgroup elements.

The paper explores properties of continuous actions on manifolds, proving bounds on subgroup size and fixed points.

Study geometric properties of a complex hyperbolic group action.

New subgroups found in CAT(0) groups with exotic finiteness properties.

The paper proves that certain spaces have injective balls of any radius.

In this article we present an unpublished proof of W. Thurston that pure braid groups have the congruence subgroup property.

The study provides obstructions and unusual subgroup properties in mapping class groups.

We prove that a dense subgroup of $\mathrm{Homeo}_{+}(I)$ is not elementary amenable. We also show that the topological group $\mathrm{Homeo}_{+}(I)$ does not satisfy the Stability of the Generators Property, moreover, any finitely generated subgroup of $\mathrm{Homeo}_{+}(I)$ admits a faithful discrete representation …

The paper studies groups with specific actions on hyperbolic spaces and finds that subgroups are either amenable or contain a free group.

We associate cube complexes called completions to each subgroup of a right-angled Coxeter group (RACG). A completion characterizes many properties of the subgroup such as whether it is quasiconvex, normal, finite-index or torsion-free. We use completions to show that reflection subgroups are quasiconvex, as are one-end…

Characterizes knotted subgroups of Lie groups and provides examples.

New subgroup found in Lie groups with unusual properties.

Polynomial growth elements found in all subgroups of Out(F_n).

We show that a finite collection of stable subgroups of a finitely generated group has finite height, finite width and bounded packing. We then use knowledge about intersections of conjugates to characterize finite families of quasimorphisms on hyperbolically embedded subgroups that can be to simultaneously extended to…

We show that any finitely generated non-elementary Kleinian group has a co-final family of finite index normal subgroups with respect to which it has Property $τ$. As a consequence, any closed hyperbolic 3-manifold has a co-final family of finite index normal subgroups for which the infimal Heegaard gradient is positiv…

Study of groups and their quasi-isometrically embedded subgroups.

Study thin hyperbolic reflection groups and their properties.

For a finitely generated group, there are two recent generalizations of the notion of a quasiconvex subgroup of a word-hyperbolic group, namely a stable subgroup and a Morse or strongly quasiconvex subgroup. Durham and Taylor defined stability and proved stability is equivalent to convex cocompactness in mapping class …

A celebrated theorem of Marshall Hall Jr. implies that finitely generated free groups are subgroup separable and that all of their finitely generated subgroups are retracts of finite-index subgroups. We use topological techniques inspired by the work of Stallings to prove that all limit groups share these two propertie…

Investigates BNSR invariants of link and knot groups, proving specific properties.

We prove a Morse Lemma for coarsely regular quasigeodesics in nonpositively curved symmetric spaces and euclidean buildings X. The main application is a simpler coarse geometric characterization of Morse subgroups of the isometry groups Isom(X) as undistorted subgroups which are coarsely uniformly regular. We show furt…

New hyperbolic groups found with specific subgroup properties.

We use partial actions, as formalized by Exel, to construct various commensurating actions. We use this in the context of groups piecewise preserving a geometric structure, and we interpret the transfixing property of these commensurating actions as the existence of a model for which the group acts preserving the geome…

New hyperbolic groups exhibit unusual finiteness properties.

The connectivity structure of graphs is typically related to the attributes of the nodes. In social networks for example, the probability of a friendship between two people depends on their attributes, such as their age, address, and hobbies. The connectivity of a graph can thus possibly be understood in terms of patte…

We study infinite covolume discrete subgroups of higher rank semisimple Lie groups, motivated by understanding basic properties of Anosov subgroups from various viewpoints (geometric, coarse geometric and dynamical). The class of Anosov subgroups constitutes a natural generalization of convex cocompact subgroups of ran…

We show that the modular group has an infinite family of finite index subgroups, each of which has the same trace set as the modular group itself. Various congruence subgroups of the modular group, and the Bianchi groups, are also shown to have this property. In the case of the modular group, we construct examples of s…

The paper studies congruence subgroups and crystallographic quotients of small Coxeter groups.

The theme of this survey is that subgroups of the mapping class group of a finite type surface S can be studied via the geometric/dynamical properties of their action on the Thurston compactification of the Teichmuller space of S, just as discrete subgroups of the isometries of hyperbolic space can be studied via their…

Each Abelian subgroup of the fundamental group of a compact and locally simply connected $d$-dimensional length space with no conjugate points is isomorphic to $\mathbb{Z}^k$ for some $0 \leq k \leq d$. It follows from this and previously known results that each solvable subgroup of the fundamental group is a Bieberbac…

Anosov subgroups generalize convex-cocompact groups in hyperbolic geometry.

We show the mapping class group, CAT(0) groups, the fundamental groups of closed 3-manifolds, and certain relatively hyperbolic groups have a local-to-global property for Morse quasi-geodesics. This allows us to generalize combination theorems of Gitik for quasiconvex subgroups of hyperbolic groups to the stable subgro…

Uniform proof of property R_infinity for specific Artin-Tits groups.

We study Veech groups of covering surfaces of primitive translation surfaces. Therefore we define congruence subgroups in Veech groups of primitive translation surfaces using their action on the homology with entries in $\mathbb{Z}/a\mathbb{Z}$. We introduce a congruence level definition and a property of a primitive t…

The study extends Dehn filling to Lie groups, ensuring geometric properties.

The study examines subgroups of RACGs and RAAGs, focusing on their RAAG properties.

The paper provides conditions for amalgamation of certain subgroups and preserves convexity properties.

We prove that for a compact subgroup $H$ of an almost connected locally compact Hausdorff group $G$, the following properties are mutually equivalent: (1) $H$ is a maximal compact subgroup of $G$, (2) $G/H$ is contractible, (3) $G/H$ is homeomorphic to a Euclidean space, (4) $G/H$ is an AE for paracompact spaces, (5) $…