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.
New proof and description of commutator subgroups for free and surface groups.
problem Understanding commutator subgroups of free and surface groups.
method Geometric proof and representation-theoretic description.
result New free generating sets and structure descriptions for commutator subgroups.
Study shows lower central subgroups of a subgroup don't contain those of the free group.
problem Whether lower central subgroups of a subgroup contain those of the free group.
method Analyzes the relationship between lower central subgroups of a free group and its subgroup.
result Lower central subgroups of a subgroup do not contain those of the free group if the subgroup does not normally generate the free group.
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.
Precise computations of Dehn functions for subgroups of free group products.
problem Computing precise Dehn functions for subgroups of direct products of free groups.
method Analyzing specific subgroups and using algebraic methods to compute Dehn functions.
result Quartic and quadratic Dehn functions for specific subgroups of free group products.
Researchers find a way to bound the complexity of certain subgroup geometric invariants.
problem Understanding the geometric invariants of subgroups of direct products of free groups.
method Generalizing techniques for 'pushing fillings' into normal subgroups.
result Finitely presented subgroups of direct products of three free groups and subgroups of finiteness type Fn−1 in a direct product of n free groups have Dehn functions bounded above by N9. We derive a generating series for the number of free subgroups of finite index in Δ+=Zp∗Zq by using a connection between free subgroups of Δ+ and certain hypermaps (also known as ribbon graphs or "fat" graphs), and show that this generating series is transcendental. We provide non-linear rec…
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.
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.
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 give several sufficient conditions for a double of a free group along a cyclic subgroup to contain a surface subgroup.
Generalizes Klein-Maskit theorem to free products of Anosov subgroups.
problem Proving a generalization of Klein-Maskit theorem for free products of Anosov subgroups.
method Analyzing Anosov subgroups in free products of semisimple Lie groups.
result The group generated by free products of Anosov subgroups is again Anosov and isomorphic to the free product.
Elementarily free groups are the finitely generated groups with the same elementary theory as free groups. We prove that elementarily free groups are subgroup separable, answering a question of Zlil Sela.
Random subgroups of free groups are invariant only by inner automorphisms.
problem Understanding the structure of random subgroups in free groups.
method Analyzing splittings and automorphisms of random subgroups.
result Random subgroups of free groups are invariant only by inner automorphisms.
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.
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…
A random graph of free groups contains a surface subgroup
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.
We prove that, except for a few explicit roots of unity, the quantum image of any Johnson subgroup of the mapping class group contains an explicit free non-abelian subgroup.
The paper proves a quantitative Tits alternative for negatively pinched manifolds.
problem Proving a quantitative version of the Tits alternative for negatively pinched manifolds.
method Analyzing discrete isometry subgroups generated by two non-elliptic isometries.
result A free subgroup of rank 2 is found in the isometry subgroup, which is convex-cocompact when one of the generators is hyperbolic.
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…
Study of Torelli subgroups of automorphism groups of free groups.
problem Understanding the structure of Torelli subgroups of Out(Fn). method Adapting techniques from 3-manifolds and surface mapping class groups.
result These groups are finitely generated and satisfy a Birman exact sequence.
Let F be a finitely generated free group. By using Bestvina-Handel theory, as well as some further improvements, the eigengroups of a given automorphism of F (and its fixed subgroup among them) are globally analyzed and described. In particular, an explicit description of all subgroups of F which occur as the fix…
New space of currents defined for nonabelian free groups and malnormal subgroups.
problem Understanding growth under iteration of outer automorphisms of nonabelian free groups.
method Introducing a new topological space of currents relative to a malnormal subgroup system.
result Currents associated with elements not in conjugates of A are dense in the space of currents relative to A. New normal subgroups found in mapping class groups.
problem Finding normal subgroups in mapping class groups.
method New windmill apparatus tailored to projection complexes.
result Normal subgroups isomorphic to non-free right-angled Artin groups.
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.
The paper explores centers of subgroups in mapping class groups and their relation to free groups and Tits alternatives.
problem Investigating centers of subgroups in mapping class groups and their properties.
method Similar techniques to show the presence of nonabelian free groups and failure of Tits alternatives.
result No big mapping class group satisfies the strong Tits alternative, and many have trivial centers.
The study characterizes subgroups of mapping tori of free groups.
problem Characterizing subgroups of mapping tori of free groups.
method Using canonical maximal sub-mapping tori and relative hyperbolicity.
result Characterizes locally quasi-convex hyperbolic groups.
The paper explores generic free subgroups and statistical hyperbolicity in group actions.
problem Understanding generic behavior of group actions and their implications.
method Analyzing statistically convex-cocompact actions and contracting elements.
result Exponential generic sets generate quasi-isometrically embedded free subgroups.
The paper finds free semigroups in dense subgroups of Lie groups with critical exponents arbitrarily close to the subgroup's.
problem Finding free semigroups with critical exponents arbitrarily close to a subgroup's in dense subgroups of Lie groups.
method Analyzing Zariski dense discrete subgroups of Lie groups, showing the existence of free semigroups with critical exponents arbitrarily close to the subgroup's.
result The existence of free semigroups with critical exponents arbitrarily close to the subgroup's in dense subgroups of Lie groups.
Algorithm determines discrete, free subgroups of SL2 over non-archimedean fields.
problem Identifying discrete, free subgroups of SL2 over non-archimedean fields.
method Ping Pong Lemma applied to Bruhat-Tits tree action.
result Algorithm determines if subgroup is discrete and free of rank two.
New Klein-Maskit theorems for Anosov subgroups.
problem Creating new Kleinian groups from old ones.
method Analogous combination theorems for Anosov subgroups.
result Construct new Kleinian groups using old ones for Anosov subgroups.
The study examines subgroups of torus mapping class group generated by Dehn twists powers.
problem Characterizing subgroups generated by powers of Dehn twists.
method Using the ping pong lemma and geometric intersection numbers.
result Subgroups can be free groups, direct products, or have specific ranks.
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.
We establish a cubic lower bound on the Dehn function of a certain finitely presented subgroup of a direct product of 3 free groups.
We show that every finitely-generated free subgroup of a right-angled, co-compact Kleinian reflection group is contained in a surface subgroup.
We study graphs of (generalized) joins and intersections of finitely generated subgroups of a free group. We show how to disprove a lemma of Imrich and Müller on these graphs and how to repair this lemma.
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.
Let H,K be two finitely generated subgroups of a free group, let ⟨H,K⟩ denote the subgroup generated by H,K, called the join of H,K, and let neither of H, K have finite index in ⟨H,K⟩. We prove the existence of an epimorphism ζ:⟨H,K⟩→F2, where F2 …
The study characterizes subgroups of braid groups and their finite quotients.
problem Understanding the relationship between braid groups and their congruence subgroups.
method Characterization and computation of finite quotients of braid groups.
result Explicit generators and free subgroups are found for certain braid groups.
A palindrome in a free group F_n is a word on some fixed free basis of F_n that reads the same backwards as forwards. The palindromic automorphism group ΠA_n of the free group F_n consists of automorphisms that take each member of some fixed free basis of F_n to a palindrome; the group ΠA_n has close connections with h…
We discuss the possibility of lifting finite subgroups, and in particular finite cyclic subgroups, with respect to the canonical projections between automorphism and outer automorphism groups of free groups, surface groups and their abelianizations.
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 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…
For random elements in free groups, we find a rank and set of subgroups.
problem Understanding the structure of subgroups containing non-primitive elements in free groups.
method Analyzing the set of subgroups of a given rank containing a non-primitive element.
result For a subset of random elements, the primitivity rank is the group rank and the set of containing subgroups is the entire group.
We prove that all atoroidal automorphisms of Out(FN) act on the space of projectivized geodesic currents with generalized north-south dynamics. As an application, we produce new examples of non virtually cyclic, free and purely atoroidal subgroups of Out(FN) such that the corresponding free group extension is hyp…
Irreducible groups cannot be free if they ergodically act on a boundary.
problem Characterizing discrete subgroups of Lie groups that act ergodically on boundaries.
method Analyzing the structure of discrete subgroups of real semi-simple Lie groups and their action on boundaries.
result Irreducible discrete subgroups of certain Lie groups cannot be free.