arXiv research

The paper disproves the existence of certain subgroups with nontrivial rational abelianization.

Finite index subgroups of certain groups cannot act faithfully on the circle.

Finite index subgroups of relatively hyperbolic groups have equal index.

The orbifold group of the Borromean rings with singular angle 90 degrees, $U$, is a universal group, because every closed oriented 3--manifold $M^{3}$ occurs as a quotient space $M^{3} = H^{3}/G$, where $G$ is a finite index subgroup of $U$. Therefore, an interesting, but quite difficult problem, is to classify the fin…

Let Mod(S) be the extended mapping class group of a surface S. For S the twice-punctured torus, we show that there exists an isomorphism of finite index subgroups of Mod(S) which is not the restriction of an inner automorphism. For S a torus with at least three punctures, we show that every injection of a finite index …

Study of subgroups in complex hyperbolic lattice triangle groups.

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 …

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…

We show that the modular group has an infinite family of finite index subgroups, each of which has the same trace set as the modular group itself. Various congruence subgroups of the modular group, and the Bianchi groups, are also shown to have this property. In the case of the modular group, we construct examples of s…

The study provides obstructions and unusual subgroup properties in mapping class groups.

Let $N$ be at least 4. We prove that every injective homomorphism from the Torelli subgroup into $Out(F_N)$ differs from the inclusion by a conjugation in $Out(F_N)$. This applies more generally to the following subgroups: every finite-index subgroup of $Out(F_N)$ (recovering a theorem of Farb and Handel); every subgro…

We associate cube complexes called completions to each subgroup of a right-angled Coxeter group (RACG). A completion characterizes many properties of the subgroup such as whether it is quasiconvex, normal, finite-index or torsion-free. We use completions to show that reflection subgroups are quasiconvex, as are one-end…

We show that any finitely generated non-elementary Kleinian group has a co-final family of finite index normal subgroups with respect to which it has Property $τ$. As a consequence, any closed hyperbolic 3-manifold has a co-final family of finite index normal subgroups for which the infimal Heegaard gradient is positiv…

Let G be a finitely generated group having the property that any action of any finite-index subgroup of G by homeomorphisms of the circle must have a finite orbit. (By a theorem of E.Ghys, lattices in simple Lie groups of real rank at least two have this property.) Suppose that such a G acts on a compact manifold M by …

Article provides conditions for embedding braid groups into mapping class groups.

Let g and n be integers at least two, and let G be the pure braid group with n strands on a closed orientable surface of genus g. We describe any injective homomorphism from a finite index subgroup of G into G. As a consequence, we show that any finite index subgroup of G is co-Hopfian.

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

We show that any isomorphism between mapping class groups of orientable infinite-type surfaces is induced by a homeomorphism between the surfaces. Our argument additionally applies to automorphisms between finite-index subgroups of these `big' mapping class groups and shows that each finite-index subgroup has finite ou…

Let S be a connected, compact and orientable surface of genus two having exactly one boundary component. We study automorphisms of the Torelli complex for S, and describe any isomorphism between finite index subgroups of the Torelli group for S. More generally, we study superinjective maps from the Torelli complex for …

We investigate the homology of finite index subgroups G_i of a given finitely presented group G. Specifically, we examine d_p(G_i), which is the dimension of the first homology of G_i, with mod p coefficients. We say that a collection of finite index subgroups {G_i} has linear growth of mod p homology if the infimum of…

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 bound the index of a subgroup in iterated Kodaira fibrations.

We show that finite index subgroups of the handlebody group are rigid in their ambient mapping class group: any injective map of a finite index subgroup of the genus $g$ handlebody group into the genus $g$ mapping class group is conjugation by a mapping class group element. On the other hand, we construct an injection …

If S is a subgroup of a direct product of two limit groups, and S is of type FP(2) over the rationals, then S has a subgroup of finite index that is a direct product of at most two limit groups.

We show that for certain arithmetic groups, geometrically finite subgroups are the intersection of finite index subgroups containing them. Examples are the Bianchi groups and the Seifert-Weber dodecahedral space. In particular, for manifolds commensurable with these groups, immersed incompressible surfaces lift to embe…

In this paper we derive a generating series for the number of cellular complexes known as pavings or three-dimensional maps, on $n$ darts, thus solving an analogue of Tutte's problem in dimension three. The generating series we derive also counts free subgroups of index $n$ in $Δ^+ = \mathbb{Z}_2*\mathbb{Z}_2*\mathbb{Z…

We study the quotient of the mapping class group $\operatorname{Mod}_g^n$ of a surface of genus $g$ with $n$ punctures, by the subgroup $\operatorname{Mod}_g^n[p]$ generated by the $p$-th powers of Dehn twists. Our first main result is that $\operatorname{Mod}_g^1 /\operatorname{Mod}_g^1[p]$ contains an infinite normal…

New insights into profinite rigidity of Kleinian groups and their subgroups.

We show that for any subgroup $H$ of Out($F_N$), either $H$ contains an atoroidal element or a finite index subgroup $H'$ of $H$ fixes a nontrivial conjugacy class in $F_N$. This result is an analog of Ivanov's subgroup theorem for mapping class groups and Handel-Mosher's subgroup theorem for Out($F_N$) in the setting …

This paper answers a basic question about the Birman exact sequence in the theory of mapping class groups. We prove that the Birman exact sequence does not admit a section over any subgroup $Γ$ contained in the Torelli group with finite index. A fortiori this proves that there is no section of the Birman exact sequence…

If F is a surface with boundary, then a finitely generated subgroup without peripheral elements of G = π_1(F) can be separated from finitely many other elements of G by a finite index subgroup of G corresponding to a finite cover F' with the same number of boundary components as F .

Paper extends Fenchel's conjecture to non-Euclidean crystallographic groups.

Let G be a finitely presented group, and let p be a prime. Then G is 'large' (respectively, 'p-large') if some normal subgroup with finite index (respectively, index a power of p) admits a non-abelian free quotient. This paper provides a variety of new methods for detecting whether G is large or p-large. These relate t…

The study shows how nonnegative Ricci curvature and metric cones imply the existence of abelian subgroups in the fundamental group of open manifolds.

A celebrated theorem of Marshall Hall Jr. implies that finitely generated free groups are subgroup separable and that all of their finitely generated subgroups are retracts of finite-index subgroups. We use topological techniques inspired by the work of Stallings to prove that all limit groups share these two propertie…

Let K be the subgroup of the extended mapping class group, Mod(S), generated by Dehn twists about separating curves. Assuming that S is a closed, orientable surface of genus at least 4, we confirm a conjecture of Farb that Comm(K), Aut(K) and Mod(S) are all isomorphic. More generally, we show that any injection of a fi…

Mapping class groups of Haken 3-manifolds enjoy many of the homological finiteness properties of mapping class groups of 2-manifolds of finite type. For example, H(M) has a torsionfree subgroup of finite index, which is geometrically finite (i. e. is the fundamental group of a finite aspherical complex). This was prove…

The paper controls the geometry of surface subgroups in specific Kleinian groups.

We prove that every endomorphism of the mapping class group of an orientable surface onto a subgroup of finite index is in fact an automorphism.

We show that all groups in a very large class of Coxeter groups are locally quasiconvex and have uniform membership problem solvable in quadratic time. If a group in the class satisfies a further hypothesis it is subgroup separable and relevant homomorphisms are also calculable in quadratic time. The algorithm also dec…

In this paper, we show that for every abelian subgroup $H$ of a Garside group, some conjugate $g^{-1}Hg$ consists of ultra summit elements and the centralizer of $H$ is a finite index subgroup of the normalizer of $H$. Combining with the results on translation numbers in Garside groups, we obtain an easy proof of the a…

If $G_1,...,G_n$ are limit groups and $S\subset G_1\times...\times G_n$ is of type $\FP_n(\mathbb Q)$ then $S$ contains a subgroup of finite index that is itself a direct product of at most $n$ limit groups. This settles a question of Sela.

A discrete subgroup of the group of isometries of the hyperbolic space is called reflective if up to a finite index it is generated by reflections in hyperplanes. The main result of this paper is a complete classification of the reflective (and quasi-reflective) subgroups among the Bianchi groups and their extensions.

The virtually cyclic dimension of Out(F_N) is finite and related properties are established.

For some $g \geq 3$, let $Γ$ be a finite index subgroup of the mapping class group of a genus $g$ surface (possibly with boundary components and punctures). An old conjecture of Ivanov says that the abelianization of $Γ$ should be finite. In this note, we prove two theorems supporting this conjecture. For the first, le…

We identify a universal group $U$ and show that $\Bbb H^3/G$ is $S^3$ when $G$ is a finite index subgroup of $U$ generated by elements of finite order.

We determine the factorial growth rate of the number of finite index subgroups of right-angled Artin groups as a function of the index. This turns out to depend solely on the independence number of the defining graph. We also make a conjecture for right-angled Coxeter groups and prove that it holds in a limited setting…

Unified algebraic framework for virtual braid structures with strong structural consequences.