arXiv research

Many 2D Artin groups are residually finite.

If $G$ is a semisimple Lie group of real rank at least 2 and $Γ$ is an irreducible lattice in $G$, then every homomorphism from $Γ$ to the outer automorphism group of a finitely generated free group has finite image.

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

We classify up to coarse equivalence all countable abelian groups of finite torsion free rank. The Q-cohomological dimension and the torsion free rank are the two invariants that give us such classification. We also prove that any countable abelian group of finite torsion free rank is coarsely equivalent to Z^n + H whe…

Solves the Wiegold problem by showing free products of left-orderable groups have normal rank > 1.

The paper proves rigidity for cocycles from higher rank lattices to Out(FN).

Study of uncountable family of finitely generated groups.

Researchers show how to perturb free group representations into higher rank groups.

Study on profinite rigidity of direct products of free and surface groups.

Suppose a group $G$ is quasi-isometric to a free product of a finite set $S$ of finitely generated abelian groups; let $S'$ denote the set of ranks of the free abelian parts of the groups in $S$. Then $G$ is commensurable with the free product of $\Z$ with a $\Z^n$ for each $n$ occurring in $S'$.

We prove that the free splitting complex of a finite rank free group, also known as Hatcher's sphere complex, is hyperbolic.

New examples show some convex-cocompact subgroups are separable.

We prove that, for n=3 and 4, the minimal nonabelian finite factor group of the outer automorphism group Out F_n of a free group of rank n is the linear group PSL_n(Z_2) (conjecturally, this may remain true for arbitrary rank n > 2). We also discuss some computational results on low index subgroups of Aut F_n and Out F…

Study of rank 2 Kleinian groups with specific parabolic elements.

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 $F_2$ …

A generalized Baumslag-Solitar (GBS) group is a finitely generated group acting on a tree with infinite cyclic edge and vertex stabilizers. We show how to determine effectively the rank (minimal cardinality of a generating set) of a GBS group; as a consequence, one can compute the rank of the mapping torus of a finite …

Carrier graphs were first introduced for closed hyperbolic 3-manifolds by White. In this paper, we first generalize this definition to carrier graphs for representations of a rank two free group into the isometry group of hyperbolic three space. Then we prove the existence and the finiteness of minimal carrier graphs f…

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

We strengthen Marshall Hall's Theorem to show that free groups are locally extended residually alternating. Let F be any free group of rank at least two, let H be a finitely generated subgroup of infinite index in F and let {g_1,...,g_n} be a finite subset of F-H. Then there is a surjection f from F to a finite alterna…

We construct nonlinear hyperbolic groups which are large, torsion-free, one-ended, and admit a finite $K(π,1)$. Our examples are built from superrigid cocompact rank one lattices via amalgamated free products and HNN extensions.

For word-equations in groups, we find a logarithmic bound on non-solutions.

We prove that the rank problem is decidable in the class of torsion-free word-hyperbolic Kleinian groups. We also show that every group in this class has only finitely many Nielsen equivalence classes of generating sets of a given cardinality.

We develop the geometry of folding paths in Outer space and, as an application, prove that the complex of free factors of a free group of finite rank is hyperbolic.

Hyperbolicity proven for a specific type of group extension.

Corrects a 1998 proof about free factors of free groups.

The paper extends a theorem to number fields without infinite places.

We extend each higher Johnson homomorphism to a crossed homomorphism from the automorphism group of a finite-rank free group to a finite-rank abelian group. We also extend each Morita homomorphism to a crossed homomorphism from the mapping class group of once-bounded surface to a finite-rank abelian group. This improve…

We discuss dense embeddings of surface groups and fully residually free groups in topological groups. We show that a compact topological group contains a nonabelian dense free group of finite rank if and only if it contains a dense surface group. Also, we obtain a characterization of those Lie groups which admit a dens…

Two groups have a common model geometry if they act properly and cocompactly by isometries on the same proper geodesic metric space. The Milnor-Schwarz lemma implies that groups with a common model geometry are quasi-isometric; however, the converse is false in general. We consider free products of uniform lattices in …

Let $G$ be the mapping torus of a polynomially growing automorphism of a finitely generated free group. We determine which epimorphisms from $G$ to $\mathbb{Z}$ have finitely generated kernel, and we compute the rank of the kernel. We thus describe all possible ways of expressing $G$ as the mapping torus of a free grou…

The study finds stably free modules and distinct 2-complexes for large ranks.

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…

We study groups acting on CAT(0) square complexes. In particular we show if Y is a nonpositively curved (in the sense of A. D. Alexandrov) finite square complex and the vertex links of Y contain no simple loop consisting of five edges, then any subgroup of the fundamental group of Y either is virtually free abelian or …

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…

New lattices in higher rank contain a fixed 3-manifold group with increasing systole.

Recall that a group is called large if it has a finite index subgroup which surjects onto a non-abelian free group. By work of Agol and Cooper-Long-Reid, most 3-manifold groups are large; in particular, the fundamental groups of hyperbolic 3-manifolds are large. In previous work, the first author gave examples of close…

We present a method for deciding when a regular abelian cover of a finite CW-complex has finite Betti numbers. To start with, we describe a natural parameter space for all regular covers of a finite CW-complex X, with group of deck transformations a fixed abelian group A, which in the case of free abelian covers of ran…

We prove that any isometry of the graph of cyclic splittings of a finitely generated free group $F_N$ of rank $N\ge 3$ is induced by an outer automorphism of $F_N$. The same statement also applies to the graphs of maximally-cyclic splittings, and of very small splittings.

We use the notion of fixity for representations of finite groups to construct free and smooth actions on products of spheres. In particular we show that a finite p-group (for p>3) will act freely and smoothly on a product of two spheres if and only if it does not contain a rank 3 elementary abelian subgroup. We show th…

Let $p$ be an odd prime. We construct a non-abelian extension $Γ$ of $S^1$ by $Z/p \times Z/p$, and prove that any finite subgroup of $Γ$ acts freely and smoothly on $S^{2p-1} \times S^{2p-1}$. In particular, for each odd prime $p$ we obtain free smooth actions of infinitely many non-metacyclic rank two $p$-groups on $…

This paper is devoted to the computation of the space $H_b^2(Γ,H;\mathbb{R})$, where $Γ$ is a free group of finite rank $n\geq 2$ and $H$ is a subgroup of finite rank. More precisely we prove that $H$ has infinite index in $Γ$ if and only if $H_b^2(Γ,H;\mathbb{R})$ is not trivial, and furthermore, if and only if there …

We give upper bounds, linear in rank, to the topological dimensions of the Gromov boundaries of the intersection graph, the free factor graph and the cyclic splitting graph of a finitely generated free group.

Let $M$ be a simply connected closed $4$-manifold. It is proved that any (possibly finite) compact Lie group acting effectively and homologically trivially on $M$ by homeomorphisms is an abelian group of rank at most two. As applications, let $\mathrm{Aut}(F_{n})$ be the automorphism group of the free group of rank $n.…

A longstanding question of Gromov asks whether every one-ended word-hyperbolic group contains a subgroup isomorphic to the fundamental group of a closed hyperbolic surface. An infinite family of word-hyperbolic groups can be obtained by taking doubles of free groups amalgamated along words that are not proper powers. W…

Study shows Morse elements are common in acylindrically hyperbolic groups.

We show that every automorphism $α$ of a free group $F_k$ of finite rank $k$ has {\it asymptotically periodic} dynamics on $F_k$ and its boundary $\partial F_k$: there exists a positive power $α^q$ such that every element of the compactum $F_k \cup \partial F_k$ converges to a fixed point under iteration of $α^q$.

Extends folding techniques to study subgroups of CAT(0) cube complexes.

A group is known as `large' if some finite index subgroup admits a surjective homomorphism onto a non-abelian free group. In this paper, we give a necessary and sufficient condition for a finitely presented group to be large, in terms of the existence of a normal series where successive quotients are finite abelian gro…