arXiv research

No hyperbolic group can have an infinite chain of free subgroups of fixed rank.

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

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

Precise computations of Dehn functions for subgroups of free group products.

Researchers find a way to bound the complexity of certain subgroup geometric invariants.

The study shows exponential distortion in virtually special groups containing free subgroups.

We derive a generating series for the number of free subgroups of finite index in $Δ^+ = \mathbb{Z}_p*\mathbb{Z}_q$ 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…

Proves existence of certain subgroups in hyperbolic groups.

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

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

The study shows subgroup separability conditions for specific groups.

Generalizes Klein-Maskit theorem to free products of Anosov subgroups.

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.

A random graph of free groups contains a surface subgroup

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…

The paper defines conditions for a free-by-free group to be hyperbolic.

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.

We show that certain classes of graphs of free groups contain surface subgroups, including groups with positive $b_2$ 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…

For a given free group $F$ of arbitrary rank (possibly infinite), and its subgroup $G$, we address the question whether a lower central subgroup of $G$ can contain a lower central subgroup of $F$. We show that the answer is no if $G$ does not normally generate $F$. The question comes from a study of Hirzebruch-type inv…

Study of Torelli subgroups of automorphism groups of free groups.

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 examples show some convex-cocompact subgroups are separable.

New space of currents defined for nonabelian free groups and malnormal subgroups.

New normal subgroups found in mapping class groups.

The study characterizes subgroups of mapping tori of free groups.

The paper finds free semigroups in dense subgroups of Lie groups with critical exponents arbitrarily close to the subgroup's.

New Klein-Maskit theorems for Anosov subgroups.

We establish a cubic lower bound on the Dehn function of a certain finitely presented subgroup of a direct product of 3 free groups.

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

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 $\langle H, K \rangle$ denote the subgroup generated by $H, K$, called the join of $H, K$, and let neither of $H$, $K$ have finite index in $\langle H, K \rangle$. We prove the existence of an epimorphism $ζ: \langle H, K \rangle \to F_2$, where $F_2$ …

The study characterizes subgroups of braid groups and their finite quotients.

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.

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…

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, ${\rm Out}(F_n)$, \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…

In this paper, we prove a quantitative version of the Tits alternative for negatively pinched manifolds $X$. Precisely, we prove that a nonelementary discrete isometry subgroup of $\mathrm{Isom}(X)$ generated by two non-elliptic isometries $g$, $f$ contains a free subgroup of rank $2$ generated by isometries $f^N , h$ …

For random elements in free groups, we find a rank and set of subgroups.

We prove that all atoroidal automorphisms of $Out(F_N)$ 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 $Out(F_N)$ such that the corresponding free group extension is hyp…

We study subgroups of the mapping class group of the torus generated by powers generated by powers of Dehn twists. We give a criterion to show when a collection of powers Dehn twists generates a free group using the ping pong lemma. We show that the subgroup generated by three uniform powers of Dehn twists can be eithe…

Irreducible groups cannot be free if they ergodically act on a boundary.

The study finds a subgroup of graph braid groups that is a direct product of non-abelian free groups.

We characterize strongly Morse quasi-geodesics in Outer space as quasi-geodesics which project to quasi-geodesics in the free factor graph. We define convex cocompact subgroups of $Out(F_n)$ as subgroups such that an orbit map in the free factor graph is a quasi-isometric embedding, and we characterize such groups via …

Combination theorems for convex projective geometry subgroups.

Geometric limits of cyclic subgroups in specific groups studied.