The study shows exponential distortion in virtually special groups containing free subgroups.
problem Understanding distortion in virtually special groups containing free subgroups.
method Constructing examples of virtually special groups with finite rank free subgroups.
result Distortion functions grow like exp^k(x^m) and can be superexponential.
Proves existence of certain subgroups in hyperbolic groups.
problem Existence of weakly malnormal quasiconvex subgroups in hyperbolic groups.
method Proof of existence in nonelementary hyperbolic groups.
result Existence of weakly malnormal, virtually free, quasiconvex subgroups in hyperbolic groups.
We show that if a group is not virtually cyclic and is hyperbolic relative to a family of proper subgroups, then it has a hyperbolically embedded subgroup which contains a finitely generated non-abelian free group as a finite index subgroup.
New result on embedding cohomology of hyperbolic groups.
problem Embedding cohomology of hyperbolic groups into their virtually free subgroups.
method Probabilistic argument and inverse limit of cohomologies.
result Second bounded cohomology of acylindrically hyperbolic groups embeds into cohomologies of virtually free subgroups.
Study shows hyperbolic subgroups can be free products of surface and free groups.
problem Characterizing hyperbolic subgroups within larger groups.
method Analyzing fiber bundles and using properties of hyperbolic groups.
result Non-elementary hyperbolic commensurated subgroups are virtually free products of surface and free groups.
We show, using Wise's equitable sets criterion, that every tubular free by cyclic group acts freely on a CAT(0) cube complex. We also show that these groups have a finite index subgroup satisfying the strongest Tits alternative, which means that every subgroup either surjects a non abelian free group or is torsion free…
The paper extends Lipschitz metric isometries between Outer Spaces of virtually free groups.
problem Extending isometry properties of Lipschitz metric to virtually free groups.
method Analyzing finite-index subgroups and their covers, identifying folding paths, and using deformation retraction.
result Existence of candidates for Lipschitz distance and deformation retraction of spine.
The study restricts Anosov subgroups of Sp(2n,R) based on subset Θ.
problem Characterizing Anosov subgroups of Sp(2n,R) based on subset Θ.
method Analyzing the structure of Anosov subgroups in terms of subset Θ.
result Anosov subgroups of Sp(2n,R) are virtually free or surface groups if Θ contains an odd integer, otherwise they are not.
The Farrell-Jones Conjecture holds for groups acting acylindrically on trees.
problem Verifying the Farrell-Jones Conjecture for groups acting on trees.
method Analyzing acylindrical actions on simplicial trees and using the Farrell-Jones Conjecture.
result The Farrell-Jones Conjecture holds for groups acting acylindrically on trees.
We prove that the profinite completion of the fundamental group of a compact 3-manifold M satisfies a Tits alternative: if a closed subgroup H does not contain a free pro-p subgroup for any p, then H is virtually soluble, and furthermore of a very particular form. In particular, the profinite completion of th…
The study establishes conditions for groups acting on polygonal complexes to contain virtually free subgroups.
problem Conditions for groups acting on polygonal complexes to contain virtually free subgroups.
method Analysis of links of polygonal complexes and conditions on their structure.
result Groups acting on polygonal complexes with certain link conditions contain virtually free subgroups.
The study shows subgroup separability conditions for specific groups.
problem Conditions for subgroup separability in free-by-cyclic and deficiency 1 groups.
method Analyzes polynomially growing monodromy and asymptotic probability of random groups.
result Random deficiency 1 groups are not subgroup separable with positive probability.
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.
New examples show some convex-cocompact subgroups are separable.
problem Whether all convex-cocompact subgroups are separable.
method Using Manning-Mj-Sageev construction, examples of separable subgroups of arbitrary finite rank are given.
result Examples of separable convex-cocompact subgroups of arbitrary finite rank exist.
We prove that if T is an R-tree with a minimal free isometric action of FN, then the Out(FN)-stabilizer of the projective class [T] is virtually cyclic. For the special case where T=T+(φ) is the forward limit tree of an atoroidal iwip element φ∈Out(FN) this is a consequence of the results o…
In this paper we generalize the notion of strongly poly-free group to a larger class of groups, we call them strongly poly-surface groups and prove that the Fibered Isomorphism Conjecture of Farrell and Jones corresponding to the stable topological pseudoisotopy functor is true for any virtually strongly poly-surface g…
Linear groups without infinite order unipotents have good properties.
problem Properties of linear groups without unipotent elements of infinite order.
method Analyzing the structure and properties of linear groups without unipotent elements of infinite order.
result Linear groups without unipotent elements of infinite order have good properties, including centralisers virtually splitting and finitely generated abelian subgroups being undistorted.
Gordon and Wilton recently proved that the double D of a free group F amalgamated along a cyclic subgroup C of F contains a surface group if a generator w of C satisfies a certain 3-manifold theoretic condition, called virtually geometricity. Wilton and the author defined the polygonality of w which also guarantees the…
New hyperbolic groups from ping-pong automorphisms.
problem Finding new hyperbolic groups from automorphisms.
method Proving generalized north-south dynamics and constructing new subgroups.
result Produced new examples of hyperbolic groups not convex cocompact.
We study subgroups of fundamental groups of real analytic closed 4-manifolds with nonpositive sectional curvature. In particular, we are interested in the following question: if a subgroup of the fundamental group is not virtually free abelian, does it contain a free group of rank two ? The technique involves the theor…
We show that certain classes of graphs of free groups contain surface subgroups, including groups with positive b2 obtained by doubling free groups along collections of subgroups, and groups obtained by "random" ascending HNN extensions of free groups. A special case is the HNN extension associated to the endomorphi…
Handlebody groups are virtual duality groups in positive genus.
problem Understanding the structure of handlebody groups.
method Analyzing mapping class groups and their dualizing modules.
result Description of dualizing modules for handlebody groups.
The study classifies subgroups of outer automorphisms of free products.
problem Classifying subgroups of outer automorphisms of free products.
method Geometric tool: boundaries of relative factor graphs and equivalence classes of arational trees.
result Every finitely generated subgroup either contains a relatively fully irreducible automorphism or virtually preserves a conjugacy class.
We give a short proof of a theorem of Handel and Mosher stating that any finitely generated subgroup of Out(FN) either contains a fully irreducible automorphism, or virtually fixes the conjugacy class of a proper free factor of FN, and we extend their result to non finitely generated subgroups of $\text{Ou…
Study virtual braid groups, proving a key subgroup result.
problem Understanding congruence subgroups in virtual braid groups.
method Using an extension of the integral Burau representation.
result Proved level 2 congruence subgroup of virtual braid group is pure virtual braid group.
Study on right-angled Coxeter groups and their geometric properties.
problem Characterizing the coarse geometry of right-angled Coxeter groups.
method Analyzing graph properties and applying geometric group theory.
result Proves properties of right-angled Coxeter groups, including quasi-isometry and divergence.
Paper proves fundamental groups of certain 3+ manifold are virtually free.
problem Understanding fundamental groups of manifolds with specific curvature properties.
method Analyzing the fundamental group of closed manifolds with (n-1)-positive Ricci curvature.
result Fundamental groups of closed manifolds with 2-positive Ricci curvature are virtually free.
The geometric dimension for proper actions gd(G) of a group G is the minimal dimension of a classifying space for proper actions EG. We construct for every integer r≥1, an example of a virtually torsion-free Gromov-hyperbolic group G such that for every group Γ which con…
A fixed point theorem is proved for inverse transducers, leading to an automata-theoretic proof of the fixed point subgroup of an endomorphism of a finitely generated virtually free group being finitely generated. If the endomorphism is uniformly continuous for the hyperbolic metric, it is proved that the set of regula…
Graphically discrete groups have strong rigidity properties.
problem Understanding the rigidity of group actions on graphs.
method Introducing graphical discreteness and proving rigidity properties.
result Free products of graphically discrete groups are action rigid.
The paper examines subgroup separability for surface and virtual braid groups.
problem Subgroup separability of surface and virtual braid groups.
method Study of subgroup separability (LERF) properties.
result Properties of subgroup separability for surface and virtual braid groups are explored.
Study of commutator subgroups and crystallographic quotients of virtual groups.
problem Investigate commutator subgroups and crystallographic quotients of virtual groups.
method Derived explicit finite presentations and proved crystallographic properties.
result Explicit finite presentations of commutator subgroups and crystallographic quotients.
We prove that if φ,ψ∈Out(FN) are hyperbolic iwips (irreducible with irreducible powers) such that <φ,ψ>≤Out(FN) is not virtually cyclic then some high powers of φ and ψ generate a free subgroup of rank two, all of whose nontrivial elements are again hyperbolic iwips. Being a hyperbolic iwip element of $…
We prove a Tits alternative theorem for groups acting on CAT(0) cubical complexes. Namely, suppose that G is a group for which there is a bound on the orders of its finite subgroups. We prove that if G acts properly on a finite-dimensional CAT(0) cubical complex, then either G contains a free subgroup of rank 2 o…
Combination theorems for convex projective geometry subgroups.
problem Understanding discrete subgroups in convex projective geometry.
method General combination theorems for discrete subgroups preserving properly convex open subsets.
result Free products of convex cocompact subgroups are convex cocompact.
We study automorphisms of a relatively hyperbolic group G. When G is one-ended, we describe Out(G) using a preferred JSJ tree over subgroups that are virtually cyclic or parabolic. In particular, when G is toral relatively hyperbolic, Out(G) is virtually built out of mapping class groups and subgroups of GL_n(Z) fixing…
Let Mod(S) denote the mapping class group of a compact, orientable surface S. We prove that finitely generated subgroups of Mod(S) which are not virtually abelian have uniform exponential growth with minimal growth rate bounded below by a constant depending only, and necessarily, on S. For the proof, we find in any suc…
We show that every virtually torsion-free subgroup of the outer automorphism group of a conjugacy separable relatively hyperbolic group is residually finite. As a direct consequence, we obtain that the outer automorphism group of a limit group is residually finite.
Abstract: Study of 2-knot groups with restrictions on normal subgroups.
problem Characterizing 2-knot groups based on their normal subgroups.
method Analyzing PD_4-complexes and using properties of π_1(X).
result Characterization of 2-knot groups based on their normal subgroups.
Unified algebraic framework for virtual braid structures with strong structural consequences.
problem Unified algebraic framework for virtual braid structures with various types of crossings.
method Introducing the universal virtual braid group UVn(c) and proving its properties. result Strong structural consequences including residual finiteness, linearity, and solvability of conjugacy problems.
We show that every virtually torsion-free subgroup of the outer automorphism group of a conjugacy separable hyperbolic group is residually finite. As a result, we are able to prove that the group of outer automorphisms of every finitely generated Fuchsian group and of every free-by-finite group is resudually finite. We…
Study shows certain infinite-ended groups are not quasi-isometrically rigid.
problem Understanding when infinite-ended groups are quasi-isometrically rigid.
method Combining results on subgroups and hyperbolic groups, adapting Whyte's argument.
result Proves certain infinite-ended groups are not quasi-isometrically rigid.
Study shows virtually abelian subgroups have commensurable counterparts in mapping class groups.
problem Understanding virtually abelian subgroups in mapping class groups.
method Proving commensurability and normalizer relationships for virtually abelian subgroups.
result Upper bounds for geometric dimension of mapping class groups for abelian subgroups of bounded rank.
Groups with similar pro-p completions have two-generated subgroups that are free.
problem Understanding the profinite rigidity of groups with specific pro-p completions. method Utilizing new results on residually-(torsion-free nilpotent) groups and their subgroups, and applying these to prove profinite rigidity.
result Groups with the same pro-p completion have two-generated subgroups that are free. Proves restrictions on projective Anosov representations of hyperbolic groups.
problem Restrictions on projective Anosov representations of hyperbolic groups.
method Word hyperbolic groups and Gromov boundary analysis.
result Word hyperbolic groups with certain properties cannot admit projective Anosov representations.
Virtual knots have similar longitude properties to classical knots.
problem Understanding longitude properties in virtual knots.
method Observation and comparison with classical knots.
result Longitudes of virtual knots lie in the second commutator subgroup of the knot group.
The paper characterizes crystallographic groups derived from virtual braid and twin groups.
problem Characterizing crystallographic groups from virtual braid and twin groups.
method Analyzing quotients of virtual braid and twin groups by their commutator subgroups.
result The quotients of virtual braid and twin groups by their commutator subgroups are crystallographic groups.
The paper extends arithmetic quotient results to right-angled Artin groups.
problem Arithmetic quotients of automorphism groups of free groups and mapping class groups.
method Analogous methods to free groups and mapping class groups applied to right-angled Artin groups.
result New virtual arithmetic quotients of Aut(F_n) for n ≥ 4, containing nonabelian free groups.