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 subgroup of Mod(S) into Mod(S) is the restriction of an inner automorphism; this completes a program begun by Irmak. For all of the above surfaces, we establish the co-Hopf property for finite index subgroups of Mod(S).
The intersection pattern of the translates of the limit set of a quasi-convex subgroup of a hyperbolic group can be coded in a natural incidence graph, which suggests connections with the splittings of the ambient group. A similar incidence graph exists for any subgroup of a group. We show that the disconnectedness of this graph for codimension one subgroups leads to splittings. We also reprove some results of Peter Kropholler on splittings of groups over malnormal subgroups and variants of them.
In a seminal paper, Stallings introduced folding of morphisms of graphs. One consequence of folding is the representation of finitely-generated subgroups of a finite-rank free group as immersions of finite graphs. Stallings's methods allow one to construct this representation algorithmically, giving effective, algorithmic answers and proofs to classical questions about subgroups of free groups. Recently Dani--Levcovitz used Stallings-like methods to study subgroups of right-angled Coxeter groups, which act geometrically on CAT(0) cube complexes. In this paper we extend their techniques to fund…
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 or $G$ is finitely generated and virtually abelian. In particular the above conclusion holds for any group $G$ with a free action on a finite-dimensional CAT(0) cubical complex.
This paper is a continuation of our previous work with Margalit where we studied group actions on projection complexes. In that paper, we demonstrated sufficient conditions so that the normal closure of a family of subgroups of vertex stabilizers is a free product of certain conjugates of these subgroups. In this paper, we study both the quotient of the projection complex by this normal subgroup and the action of the quotient group on the quotient of the projection complex. We show that under certain conditions that the quotient complex is $δ$-hyperbolic. Additionally, under certain circumstan…
For every finite graph $Γ$, we define a simplicial complex associated to the outer automorphism group of the RAAG $A_Γ$. These complexes are defined as coset complexes of parabolic subgroups of $Out^0(A_Γ)$ and interpolate between Tits buildings and free factor complexes. We show that each of these complexes is homotopy Cohen-Macaulay and in particular homotopy equivalent to a wedge of d-spheres. The dimension d can be read off from the defining graph $Γ$ and is determined by the rank of a certain Coxeter subgroup of $Out^0(A_Γ)$. In order to show this, we refine the decomposition sequence for…
Let $S$ be a closed, connected, orientable surface of genus at least 3, $\mathcal{C}(S)$ be the complex of curves on $S$ and $Mod_S^*$ be the extended mapping class group of $S$. We prove that a simplicial map, $λ: \mathcal{C}(S) \to \mathcal{C}(S)$, preserves nondisjointness (i.e. if $α$ and $β$ are two vertices in $\mathcal{C}(S)$ and $i(α, β) \neq 0$, then $i(λ(α), λ(β)) \neq 0$) iff it is induced by a homeomorphism of $S$. As a corollary, we prove that if $K$ is a finite index subgroup of $Mod_S^*$ and $f: K \to Mod_S^*$ is an injective homomorphism, then $f$ is induced by a homeomorphism …
Let $g\geq3$ and $n\geq0$, and let ${\mathcal{M}}_{g,n}$ be the mapping class group of a surface of genus $g$ with $n$ boundary components. We prove that ${\mathcal{M}}_{g,n}$ contains a unique subgroup of index $2^{g-1}(2^{g}-1)$ up to conjugation, a unique subgroup of index $2^{g-1}(2^{g}+1)$ up to conjugation, and the other proper subgroups of ${\mathcal{M}}_{g,n}$ are of index greater than $2^{g-1}(2^{g}+1)$. In particular, the minimum index for a proper subgroup of ${\mathcal{M}}_{g,n}$ is $2^{g-1}(2^{g}-1)$.
A square complex is a 2-complex formed by gluing squares together. This article is concerned with the fundamental group $Γ$ of certain square complexes of nonpositive curvature, related to quaternion algebras. The abelian subgroup structure of $Γ$ is studied in some detail.
We develop the foundations of the theory of relatively geometric actions of relatively hyperbolic groups on CAT(0) cube complexes, a notion introduced in our previous work [5]. In the relatively geometric setting we prove: full relatively quasi-convex subgroups are convex compact; an analog of Agol's Theorem; and a version of Haglund--Wise's Canonical Completion and Retraction.
A group is coherent if all its finitely generated subgroups are finitely presented. In this article we provide a criterion for positively determining the coherence of a group. This criterion is based upon the notion of the perimeter of a map between two finite 2-complexes which is introduced here. In the groups to which this theory applies, a presentation for a finitely generated subgroup can be computed in quadratic time relative to the sum of the lengths of the generators. For many of these groups we can show in addition that they are locally quasiconvex. As an application of these results w…
We define an integer-valued invariant of special cube complexes called the genus, and prove that having genus one characterizes special cube complexes with abelian fundamental group. Using the genus, we obtain a new proof that the fundamental group of a special cube complex is either free abelian or surjects onto a non-cyclic free group. We also investigate automorphisms of special cube complexes, and give a new geometric proof that the Torelli subgroup for a right-angled Artin group is torsion-free.
Let R be a compact, connected, orientable surface of genus g with p boundary components. Let C(R) be the complex of curves on R and Mod_R^* be the extended mapping class group of R. Suppose that either g = 2 and p > 1 or g > 2 and p >= 0. We prove that a simplicial map lambda from C(R) to C(R) is superinjective if and only if it is induced by a homeomorphism of R. As a corollary, we prove that if K is a finite index subgroup of Mod_R^* and f is an injective homomorphism from K to Mod_R^*, then f is induced by a homeomorphism of R and f has a unique extension to an automorphism of Mod_R^*. This…
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 order outer automorphism of a free group $F_n$. We also show that the rank of a finite index subgroup of a GBS group G cannot be smaller than the rank of G. We determine which GBS groups are large (some finite index subgroup maps onto $F_2$), and we solve the commensurability pro…
Combinatorial aspects of the Torelli-Johnson-Morita theory of surface automorphisms are extended to certain subgroups of the mapping class groups. These subgroups are defined relative to a specified homomorphism from the fundamental group of the surface onto an arbitrary group $K$. For $K$ abelian, there is a combinatorial theory akin to the classical case, for example, providing an explicit cocycle representing the first Johnson homomophism with target $Λ^3 K$. Furthermore, the Earle class with coefficients in $K$ is represented by an explicit cocyle.
We prove that the conjugacy problem in right-angled Artin groups (RAAGs), as well as in a large and natural class of subgroups of RAAGs, can be solved in linear-time. This class of subgroups contains, for instance, all graph braid groups (i.e. fundamental groups of configuration spaces of points in graphs), many hyperbolic groups, and it coincides with the class of fundamental groups of ``special cube complexes'' studied independently by Haglund and Wise.
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 contains a free group of rank two. In addition we discuss when a group generated by two hyperbolic isometries contains a free group of rank two and when two points in the ideal boundary of a CAT(0) 2-complex at Tits distance $π$ apart are the endpoints of a geodesic in the 2-comp…
We provide a new method of constructing non-quasiconvex subgroups of hyperbolic groups by utilizing techniques inspired by Stallings' foldings. The hyperbolic groups constructed are in the natural class of right-angled Coxeter groups (RACGs for short) and can be chosen to be 2-dimensional. More specifically, given a non-quasiconvex subgroup of a (possibly non-hyperbolic) RACG, our construction gives a corresponding non-quasiconvex subgroup of a hyperbolic RACG. We use this to construct explicit examples of non-quasiconvex subgroups of hyperbolic RACGs including subgroups whose generators are a…
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-ended Coxeter subgroups of a 2-dimensional RACG. We provide an algorithm that determines whether a given one-ended, 2-dimensional RACG is isomorphic to some finite-index subgroup of another given RACG. In addition, we answer several algorithmic questions regarding quasiconvex subgro…
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, let $T_x$ denote the Dehn twist about a simple closed curve $x$. For some $n \geq 1$, we have $T_x^n \in Γ$. We prove that $T_x^n$ is torsion in the abelianization of $Γ$. Our second result shows that the abelianization of $Γ$ is finite if $Γ$ contains a "large chunk" (in a certain…
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.
Let $M(N_{h,n})$ denote the mapping class group of a compact nonorientable surface of genus $h\ge 7$ and $n\le 1$ boundary components, and let $T(N_{h,n})$ be the subgroup of $M(N_{h,n})$ generated by all Dehn twists. It is known that $T(N_{h,n})$ is the unique subgroup of $M(N_{h,n})$ of index $2$. We prove that $T(N_{h,n})$ (and also $M(N_{h,n})$) contains a unique subgroup of index $2^{g-1}(2^g-1)$ up to conjugation, and a unique subgroup of index $2^{g-1}(2^g+1)$ up to conjugation, where $g=\lfloor(h-1)/2\rfloor$. The other proper subgroups of $T(N_{h,n})$ and $M(N_{h,n})$ have index great…
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 finite index subgroups of the universal group $U$. One of the purposes of this paper is to begin this classification. In particular we analyze the classification of the finite index subgroups of $U$ that are generated by rotations.
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 recurrence relations for the above numbers based on differential equations that are part of the Riccati hierarchy. We also study the generating series for conjugacy classes of free subgroups of finite index in $Δ^+$, which correspond to isomorphism classes of hypermaps. Asymptotic f…
Let $Σ_{g}$ be a closed, connected, and oriented surface of genus $g \geq 24$ and let $Γ$ be a finite index subgroup of the mapping class group $Mod(Σ_{g})$ that contains the Torelli group $\mathcal{I}(Σ_g)$. Then any orientation preserving $C^1$ action of $Γ$ on the circle cannot be faithful. We also show that if $Γ$ is a finite index subgroup of $Aut(F_n)$, when $n \geq 8$, that contains the subgroup of IA-automorphisms, then any orientation preserving $C^1$ action of $Γ$ on the circle cannot be faithful. Similarly, if $Γ$ is a finite index subgroup of $Out(F_n)$, when $n \geq 8$, that conta…
We say a $d$-dimensional simplicial complex embeds into double dimension if it embeds into the Euclidean space of dimension $2d$. For instance, a graph is planar iff it embeds into double dimension. We study the conditions under which the join of two simplicial complexes embeds into double dimension. Quite unexpectedly, we show that there exist complexes which do not embed into double dimension, however their join embeds into the respective double dimension. We further derive conditions, in terms of the van Kampen obstructions of the two complexes, under which the join will not be embeddable i…
Ivanov's finite-index Frattini conjecture for subgroups of mapping class groups
Université Bourgogne Europe, CNRS, IMB UMR 5584（勃艮第欧洲大学）；Institut Universitaire de France（法国高等研究院）；Department of Mathematics, Michigan State University（密歇根州立大学数学系）
In this paper we show that the cohomology of a connected CW complex is periodic if and only if it is the base space of an orientable spherical fibration with total space that is homotopically finite dimensional. As applications we characterize those discrete groups that act freely and properly on a cartesian product of euclidean space and a sphere; we construct non-standard free actions of rank two simple groups on finite complexes Y homotopy equivalent to a product of two spheres and we prove that a finite p-group P acts freely on such a complex if and only if it does not contain a subgroup i…
For a given $ε>0$, we show that there exist two finite index subgroups of $PSL_2(\mathbb{Z})$ which are $(1+ε)$-quasisymmetrically conjugated and the conjugation homeomorphism is not conformal. This implies that for any $ε>0$ there are two finite regular covers of the Modular once punctured torus $T_0$ (or just the Modular torus) and a $(1+ε)$-quasiconformal between them that is not homotopic to a conformal map. As an application of the above results, we show that the orbit of the basepoint in the Teichmüller space $T(§)$ of the punctured solenoid $§$ under the action of the corresponding Modu…
