New method constructs non-quasiconvex subgroups in hyperbolic groups.
problem Creating non-quasiconvex subgroups in hyperbolic groups.
method Using Stallings-like techniques on right-angled Coxeter groups (RACGs).
result Explicit examples of non-quasiconvex subgroups constructed.
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.
Proves congruence subgroup property for mapping class groups of hyperbolic surfaces.
problem Residual finiteness of hyperbolic groups and congruence subgroup property for mapping class groups.
method Assumption of residual finiteness of hyperbolic groups leads to proof of congruence subgroup property.
result Congruence subgroup property for mapping class groups of hyperbolic surfaces.
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.
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 FP2 but not finitely presented. We give uncountably many groups of type FP2 with similar properties to those subgro…
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.
No hyperbolic group can have an infinite chain of free subgroups of fixed rank.
problem Infinite ascending chains of free subgroups in hyperbolic groups.
method Proof by contradiction and properties of hyperbolic groups.
result Hyperbolic groups do not contain strictly ascending chains of free quasiconvex subgroups of constant rank.
Suppose that all hyperbolic groups are residually finite. The following statements follow: In relatively hyperbolic groups with peripheral structures consisting of finitely generated nilpotent subgroups, quasiconvex subgroups are separable; Geometrically finite subgroups of non-uniform lattices in rank one symmetric sp…
Mahan Mitra (Mj) proved Cannon--Thurston maps exist for normal hyperbolic subgroups of a hyperbolic group. We prove that Cannon--Thurston maps do not exist for infinite normal hyperbolic subgroups of non-hyperbolic CAT(0) groups with isolated flats with respect to the visual boundaries. We also show Cannon--Thurston ma…
Finite index subgroups of relatively hyperbolic groups have equal index.
problem Finite index subgroups of relatively hyperbolic groups have equal index.
method Demonstrating that the number of simplices in a simplicial classifying space grows linearly with index.
result Finite index subgroups of relatively hyperbolic groups have equal index.
Uniform undistortion in cyclic subgroups of certain groups.
problem Understanding undistorted subgroups in group actions.
method Using quasimorphisms and hierarchically hyperbolic groups.
result Sharp examples of undistorted subgroups in hierarchically hyperbolic groups.
Study of subgroups in complex hyperbolic lattice triangle groups.
problem Characterizing subgroups of finite index in complex hyperbolic lattice triangle groups.
method Explicit construction and analysis of subgroups, examination of their properties.
result Identification of neat subgroups, subgroups with positive first Betti number, and homomorphisms onto non-Abelian free groups.
New hyperbolic groups found with specific subgroup properties.
problem Finding hyperbolic groups with certain subgroup types.
method Dehn filling from a specific lattice.
result Groups containing subgroups of type F3 but not F4. Explicit polynomial bound found for subgroup Dehn function.
problem Finding explicit bounds on Dehn functions of subgroups of hyperbolic groups.
method Constructing a specific example of a non-hyperbolic subgroup and analyzing its Dehn function.
result Explicit polynomial upper bound n96 on the Dehn function of a non-hyperbolic subgroup. Classifies hyperbolic groups with surface-like boundaries.
problem Classifying hyperbolic groups with specific surface-like boundaries.
method Analyzing quasiconvex codimension-1 surface subgroups with trivial or cyclic intersections.
result Identifies hyperbolic groups with surface-like boundaries.
Study complex hyperbolic lattices and their subgroups, proving new finiteness properties.
problem Characterize subgroups of complex hyperbolic lattices.
method Analyzing homomorphisms and using arithmetic lattice properties.
result Deep subgroups of complex hyperbolic lattices admit homomorphisms to Z with specific kernel types.
The study finds a limit on subgroup complexity in hyperbolic 3-manifold groups.
problem Understanding subgroups of bounded rank in hyperbolic 3-manifold groups.
method Proving a finiteness theorem for subgroups of bounded rank.
result Every bounded rank covering tower of closed hyperbolic 3-manifolds is a tower of finite covers associated to a fibration over a 1-orbifold.
The study finds that certain hyperbolic manifolds contain subgroups isomorphic to surface groups.
problem The existence of thin surface subgroups in non-uniform arithmetic lattices.
method Analyzes arithmetic hyperbolic manifolds and their fundamental groups.
result Fundamental groups of non-compact arithmetic hyperbolic manifolds contain thin surface subgroups.
Study of groups and their quasi-isometrically embedded subgroups.
problem Understanding the structure and properties of groups and their subgroups.
method Abstracting the notion of A/QI triples and using methods from geometric group theory.
result Stability of quasi-isometrically embedded subgroups in finitely generated groups.
Study thin hyperbolic reflection groups and their properties.
problem Characterize and enumerate thin hyperbolic reflection groups.
method Analyze Zariski dense subgroups of hyperbolic isometries, apply Vinberg algorithm.
result All thin hyperbolic reflection groups are enumerable.
Let G be an acylindrically hyperbolic group. We consider a random subgroup H in G, generated by a finite collection of independent random walks. We show that, with asymptotic probability one, such a random subgroup H of G is a free group, and the semidirect product of H acting on E(G) is hyperbolically embedded in G, w…
The notions of stable and Morse subgroups of finitely generated groups generalize the concept of a quasiconvex subgroup of a word-hyperbolic group. For a word-hyperbolic group G, Kapovich provided a partial algorithm which, on input a finite set S of G, halts if S generates a quasiconvex subgroup of G and run…
Criterion for congruence RFRS towers in hyperbolic lattices.
problem Criteria for hyperbolic lattices to have congruence RFRS towers.
method Criterion based on congruence subgroups.
result Virtually fibered Bianchi groups and new RFRS Kähler groups.
For relatively hyperbolic groups, we investigate conditions guaranteeing that the subgroup generated by two relatively quasiconvex subgroups Q1 and Q2 is relatively quasiconvex and isomorphic to Q1∗Q1∩Q2Q2. The main theorem extends results for quasiconvex subgroups of word-hyperbolic groups, an…
The paper characterizes convex co-compact groups with one-dimensional boundary faces.
problem Characterizing convex co-compact groups with specific boundary properties.
method Proving relative hyperbolicity and using coarse Hilbert dimension.
result Convex co-compact groups with one-dimensional boundary faces are relatively hyperbolic.
New theorem bounds group quotient size to subgroups index.
problem Understanding subgroup structure in hyperbolic groups.
method Proved quotient size bounds on Eilenberg-MacLane spaces.
result One-ended hyperbolic groups cannot have isomorphic finite-index subgroups.
We classify the 3-dimensional hyperbolic polyhedral orbifolds that contain no embedded essential 2-suborbifolds, up to decomposition along embedded hyperbolic triangle orbifolds (turnovers). We give a necessary condition for a 3-dimensional hyperbolic polyhedral orbifold to contain an immersed (singular) hyperbolic tur…
Locally connected boundaries proven for relatively hyperbolic groups.
problem Proving local connectedness of boundaries for relatively hyperbolic groups.
method Using a group pair (Γ,P) that is relatively one ended, and removing restrictions on cardinality and peripheral subgroups. result The Bowditch boundary of (Γ,P) is locally connected. Anosov subgroups generalize convex-cocompact groups in hyperbolic geometry.
problem Understanding convex-cocompact subgroups in higher rank geometry.
method Characterizing Anosov subgroups and comparing them to convex-cocompact groups.
result Anosov subgroups are the right generalizations of convex-cocompact groups in hyperbolic geometry.
In this paper, we state two combination theorems for relatively quasiconvex subgroups in a relatively hyperbolic group. Applications are given to the separability of double cosets of certain relatively quasiconvex subgroups and the existence of closed surface subgroups in relatively hyperbolic groups.
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≥5. We construct another eight "forbidden" grap…
Extends growth properties of hyperbolic groups to their extensions.
problem Quantifying subgroup alternatives in group laws.
method Develops a framework for preserving exponential growth in extensions of hyperbolic groups.
result Automorphism groups of certain hyperbolic and Artin groups have locally uniform exponential growth.
The paper classifies fixed subgroups of endomorphisms in free-abelian times surface groups.
problem Characterizing fixed subgroups of endomorphisms in specific group structures.
method Study of endomorphisms, classification of fixed subgroups, and equivalent conditions for end-fixed subgroups.
result Complete classification of fixed subgroups in free-abelian times surface groups.
The paper defines conditions for a free-by-free group to be hyperbolic.
problem Conditions for a free-by-free group to be relatively hyperbolic.
method Necessary and sufficient conditions involving exponentially growing automorphisms and invariant subgroup systems.
result A subgroup system can be used to construct peripheral subgroups making the extension hyperbolic.
We show that, if H is a random subgroup of a finitely generated free group Fk, only inner automorphisms of Fk may leave H invariant. A similar result holds for random subgroups of toral relatively hyperbolic groups, more generally of groups which are hyperbolic relative to slender subgroups. These results fol…
Stable subgroups identified in genus two handlebody group.
problem Characterizing stable subgroups in genus two handlebody group.
method Proving genus two handlebody group is hierarchically hyperbolic, using quasi-isometric embedding properties and Hamenstädt-Hensel construction.
result Stable subgroups identified and characterized.
Using a probabilistic argument we show that the second bounded cohomology of an acylindrically hyperbolic group G (e.g., a non-elementary hyperbolic or relatively hyperbolic group, non-exceptional mapping class group, Out(Fn), \dots) embeds via the natural restriction maps into the inverse limit of the secon…
We propose several common extensions of the classes of Anosov subgroups and geometrically finite Kleinian groups among discrete subgroups of semisimple Lie groups. We relativize various dynamical and coarse geometric characterizations of Anosov subgroups given in our earlier work, extending the class from intrinsically…
Generalized Agol's theorem to 3-manifold groups.
problem Two-parabolic-generator subgroups in hyperbolic 3-manifold groups.
method Generalization of Agol's proof for 2-bridge link groups to 3-manifold groups.
result Refinement of Boileau-Weidmann's result.
The paper examines random walks on metric spaces and finds commensurable subgroups.
problem Determining commensurable subgroups via stationary measures in metric spaces.
method Analyzing random walks on isometry groups of metric spaces with non-singular stationary measures.
result Subgroups generated by random walks are commensurable under mild conditions.
We introduce and study the notions of hyperbolically embedded and very rotating families of subgroups. The former notion can be thought of as a generalization of the peripheral structure of a relatively hyperbolic group, while the later one provides a natural framework for developing a geometric version of small cancel…
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…
Study geometric properties of a complex hyperbolic group action.
problem Geometric properties of a specific modular group action.
method Explicit description of subgroups and conjugacy classes.
result Explicit description of a torsion-free subgroup of index 336.
We obtain a number of finiteness results for groups acting on Gromov-hyperbolic spaces. In particular we show that a torsion-free locally quasiconvex hyperbolic group has only finitely many conjugacy classes of n-generated one-ended subgroups. We also show that the rank problem is solvable for the class of torsion-fr…
The paper extends quasimorphisms on subgroups to larger groups.
problem Extending quasimorphisms from subgroups to larger groups.
method Provides a general sufficient condition for extendability of quasimorphisms on subgroups.
result New results for quasimorphisms on normal subgroups, including bi-Lipschitz equivalence of stable commutator length and group-theoretic Dehn filling.
The study restricts normal subgroups of Kähler groups, proving specific cases and general restrictions.
problem Characterizing normal subgroups of Kähler groups.
method Analyzing embeddings and conjugation actions of surface groups and one-ended hyperbolic groups.
result Restrictions on normal subgroups of Kähler groups, including virtual direct products and surface group properties.
New groups algebraically fibre with high-dimensional hyperbolic groups.
problem Finding new quasi-isometry classes of hyperbolic groups.
method Constructing infinitely many hyperbolic groups as finite-index subgroups of right-angled Coxeter groups.
result Groups algebraically fibre with finitely presented kernels, expanding finiteness 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 …