Academic literature on the topic 'Mapping class subgroups'

We construct the first examples of normal subgroups of mapping class groups that are isomorphic to non-free right-angled Artin groups. Our construction also gives normal, non-free right-angled Artin subgroups of other groups, such as braid groups and pure braid groups, as well as many subgroups of the mapping class group, such as the Torelli subgroup. Our work recovers and generalizes the seminal result of Dahmani–Guirardel–Osin, which gives free, purely pseudo-Anosov normal subgroups of mapping class groups. We give two applications of our methods: (1) we produce an explicit proper normal subgroup of the mapping class group that is not contained in any level $m$ congruence subgroup and (2) we produce an explicit example of a pseudo-Anosov mapping class with the property that all of its even powers have free normal closure and its odd powers normally generate the entire mapping class group. The technical theorem at the heart of our work is a new version of the windmill apparatus of Dahmani–Guirardel–Osin, which is tailored to the setting of group actions on the projection complexes of Bestvina–Bromberg–Fujiwara.

Let S S be a surface and let Mod ⁡ ( S , K ) \operatorname {Mod}(S,K) be the mapping class group of S S permuting a Cantor subset K ⊂ S K \subset S . We prove two structure theorems for normal subgroups of Mod ⁡ ( S , K ) \operatorname {Mod}(S,K) . (Purity:) if S S has finite type, every normal subgroup of Mod ⁡ ( S , K ) \operatorname {Mod}(S,K) either contains the kernel of the forgetful map to the mapping class group of S S , or it is ‘pure’ — i.e. it fixes the Cantor set pointwise. (Inertia:) for any n n element subset Q Q of the Cantor set, there is a forgetful map from the pure subgroup PMod ⁡ ( S , K ) \operatorname {PMod}(S,K) of Mod ⁡ ( S , K ) \operatorname {Mod}(S,K) to the mapping class group of ( S , Q ) (S,Q) fixing Q Q pointwise. If N N is a normal subgroup of Mod ⁡ ( S , K ) \operatorname {Mod}(S,K) contained in PMod ⁡ ( S , K ) \operatorname {PMod}(S,K) , its image N Q N_Q is likewise normal. We characterize exactly which finite-type normal subgroups N Q N_Q arise this way. Several applications and numerous examples are also given.

For a finitely generated group, there are two recent generalizations of the notion of a quasiconvex subgroup of a word-hyperbolic group, namely a stable subgroup and a Morse or strongly quasiconvex subgroup. Durham and Taylor [M. Durham and S. Taylor, Convex cocompactness and stability in mapping class groups, Algebr. Geom. Topol. 15(5) (2015) 2839–2859] defined stability and proved stability is equivalent to convex cocompactness in mapping class groups. Another natural generalization of quasiconvexity is given by the notion of a Morse or strongly quasiconvex subgroup of a finitely generated group, studied recently by Tran [H. Tran, On strongly quasiconvex subgroups, To Appear in Geom. Topol., preprint (2017), arXiv:1707.05581 ] and Genevois [A. Genevois, Hyperbolicities in CAT (0) cube complexes, preprint (2017), arXiv:1709.08843 ]. In general, a subgroup is stable if and only if the subgroup is Morse and hyperbolic. In this paper, we prove that two properties of being Morse and stable coincide for a subgroup of infinite index in the mapping class group of an oriented, connected, finite type surface with negative Euler characteristic.

Abstract 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 outer automorphism group. As a key ingredient, we prove that all simplicial automorphisms between curve complexes of infinite-type orientable surfaces are induced by homeomorphisms.

This thesis studies the subgroup structure of mapping class groups. We use techniques that fall into two categories: analysing the group action on a family of simplicial complexes, and investigating regular, finite-sheeted covering spaces. We use the first approach to prove that a wide class of normal subgroups of mapping class groups of punctured surfaces are geometric, that is, they have the extended mapping class group as their group of automorphisms, expanding on work of BrendleMargalit. For example, we determine that every member of the Johnson filtration is geometric. By considering punctured spheres, we also establish the automorphism groups of many normal subgroups of the braid group. The second approach is to relate subgroups of each of the mapping class groups associated to a covering space, namely, the liftable and symmetric mapping class groups. Given that the two surfaces have boundary, we consider covers in which either every mapping class lifts or every mapping class is fibre-preserving. We classify all covers that fall into one of these cases. In Chapter 1 we recall some preliminaries before stating the main results of the thesis. We then extend Brendle-Margalit's definition of complexes of regions to surfaces with punctures. Chapter 2 proves that the automorphism group of a complex of regions is the extended mapping class group, resolving in part a metaconjecture of N. V. Ivanov. In Chapter 3 we construct a complex of regions associated to a general normal subgroup of a mapping class group of a surface with punctures. We then apply the main result of the previous chapter to establish that such a normal subgroup is geometric. Finally, Chapter 4 presents joint work with Tyrone Ghaswala. We give a proof of the Birman-Hilden Theorem for surfaces with boundary and then prove the classifications of regular, finite-sheeted covering spaces of surfaces with boundary discussed above. We conclude by investigating an infinite family of branched covers of the disc. This family induces embeddings of the braid group into mapping class groups. We prove that each of these embeddings maps a standard generator of the braid group to a product of Dehn twists about curves forming a chain, providing an answer to a question of Wajnryb.

Given a convex cocompact subgroup G < Mod(S), and points x, y ∈ Teich(S) we obtain asymptotic formulas as R → ∞ 1 of |B_R(|x ) [special character omitted] G_y| as well as the number of conjugacy classes of pseudo-Anosov elements in G of dilatation at most R. We do this by developing an analogue of Patterson-Sullivan theory for the action of G on PMF.

Dans cette thèse, nous nous intéressons à la dynamique de sous-groupes modulaires sur la variété des U(2)- caractères . Plus précisément, nous étudions des questions d'ergodicité de l'action de sous-groupes G du groupe modulaire Mod(g,n) d'une surface compacte S(g,n) de genre g et n composantes de bord. Ces questions ont été naturellement posées après la preuve de Goldman de l'ergodicité du groupe modulaire sur la variété des caractères. Le premier résultat général dans cette direction est dû à Funar et Marché, en montrant que le premier sous-groupe de Johnson agit de manière ergodique sur la variété des caractères, pour toute surface fermée S(g). D'un autre coté, Brown a montré l'existence de points fixes elliptiques pour tout sous-groupe généré par un homéomorphimse pseudo-Anosov sur le tore épointé S(1,1). Ceci a permis de démontrer la non-ergodicité de tels sous-groupes par Forni, Goldman, Lawton et Matheus en appliquant la théorie KAM. Dans la première partie de la thèse, nous étudions une dynamique naturelle sur l'espace des modules des triangles sphériques de la sphère de dimension 2 en reliant cette dynamique à la dynamique du groupe modulaire SL(2, Z) sur la variété des caractères du tore épointé. La deuxième partie est consacrée à l'étude de l'existence de points fixes elliptiques pour les homéo\-morphismes pseudo-Anosov sur les variétés de caractères des surfaces épointée S(g,n), où g est égal à 0 ou 1. On montre que dans le cas de la variété des caractères relative correspondant à un niveau k du tore épointé, pour un ensemble de mesure positive et dense de niveaux de la fonction invariante k, il existe une famille d'élements pseudo-Anosov qui n'agissent pas érgodiquement sur ces niveaux. dans le cas du tore épointé S(1,1). Un résultat similaire est démontré pour un ensemble de paramètres B dans le cas de la sphère à quatre trous. Ces résultats sont peuvent être combinés pour construire une famille d'éléments pseudo-Anosov sur le tore à deux trous S(1,2), qui admettent un point fixe elliptique. Nous discutons ensuite de l'action d'un groupe G généré par des twists de Dehn le long d'une paire de multi-courbes qui remplissent la surface ou plus généralement le long d'une famille des courbes qui remplissent S(g). Nous montrons dans cette partie qu'il existe deux multi-courbes qui remplissent la surface de genre deux S(2) dont les twists de Dehn associées génèrent un groupe G agissant de manière non-ergodique sur la variété des representations, en trouvant des fonctions rationnelles invariantes explicites. De même, nous montrons l’existence de fonctions rationnelles invariantes par conjugaison et invariantes par un sous-groupe G générées par des twists de Dehn le long d'une famille des courbes qui remplissent la surface fermée non-orientable de genre 4
In this thesis, we are interested in the dynamics of the mapping class subgroups on the U(2) character variety. More precisely, we deal with ergodicity questions of a subgroup G of the mapping class group Mod(g,n) of a compact surface S(g,n) of genus g and n boundary components. These questions were naturally raised after Goldman's proof of the ergodicity of mapping class groups on the SU(2)-character variety. The first general result in this direction is due to Funar and Marché by showing that the first Johnson subgroups act ergodically on the character variety, for any closed surfaces S(g). On the other hand, Brown showed the existence of an elliptic fixed point (or a double elliptic fixed point) for any subgroup generated by a pseudo-Anosov element on the punctured torus S(1,1). This led to the proof of the non-ergodicity of such subgroups by Forni, Goldman, Lawton, and Mateus by applying KAM theory. In the first part of the thesis, we study the natural dynamics of the moduli space of spherical triangles on the 2-sphere relating these dynamics to the dynamics of the mapping class group on the SU(2)-character variety of the punctured torus.The second part is devoted to the study of the existence of elliptic fixed points for pseudo-Anosov homeomorphisms on the character varieties of punctured surfaces S(g,n), where g is 0 or 1. By showing that near any relative character variety of the once punctured torus, for a set of positive measure and dense of levels k, there exists a family of pseudo-Anosov elements that do not act ergodically on that level, in the case of the punctured torus S(1,1). A similar result holds for a set of parameters B in the case of the four-punctured sphere S(0,4). Then these results can be combined to construct a family of pseudo-Anosov elements on the twice-punctured torus S(1,2) that admit an elliptic fixed point.We discuss then the action of a group G generated by Dehn-twist along a pair of filling multi-curves or along a family of filling curves on S(g). We show in this part that there exist two filling multi-curves on the surface of genus two S(2) whose associated Dehn twists generate a group G acting non-ergodically on representation variety by finding explicit invariant rational functions. Similarly, We found invariant rational functions of a subgroup G generated by Dehn-twists along a family of filling loops on the character variety of the non-orientable surface of genus 4

Dans la première partie de cette thèse on étudiera la conjecture de généricité: dans le graphe de Cayley du groupe modulaire d'une surface fermée on regarde une boule centrée à l'identité et on s'intéresse à la proportion de sommets pseudo-Anosov dans cette boule. La conjecture de généricité affirme que cette proportion doit tendre vers 1 quand le rayon de la boule tend vers l'infini. On montre qu'elle est bornée inférieurement par un nombre strictement positif et on montre des résultats similaires pour une grande classe de sous-groupes du groupe modulaire. On présente aussi des résultats analogues pour des groupes d'Artin-Tits de type sphérique, en sachant que dans ce cas, être pseudo-Anosov est analogue à agir loxodromiquement sur un complexe delta-hyperbolique convenable. Dans la deuxième partie on donne des résultats sur les sous-groupes paraboliques des groupes d'Artin-Tits de type sphérique: le standardisateur minimal d'une courbe dans le disque troué est la tresse minimale positive qui la fait devenir ronde. On construit un algorithme pour le calculer d'une façon géométrique. Ensuite, on généralise le problème pour les groupes d'Artin-Tits de type sphérique. On montre aussi que l'intersection de deux sous-groupes paraboliques est un sous-groupe parabolique et que l'ensemble de sous-groupes paraboliques est un treillis par rapport à l'inclusion. Finalement, on définit le complexe simplicial des sous-groupes paraboliques irréductibles, et on le propose comme l'analogue du complexe de courbes
In the first part of this thesis we study the genericity conjecture: In the Cayley graph of the mapping class group of a closed surface we look at a ball of large radius centered on the identity vertex, and at the proportion of pseudo-Anosov vertices among the vertices in this ball. The genericity conjecture states that this proportion should tend to one as the radius tends to infinity. We prove that it stays bounded away from zero and prove similar results for a large class of subgroups of the mapping class group. We also present analogous results for Artin--Tits groups of spherical type, knowing that in this case being pseudo-Anosov is analogous to being a loxodromically acting element. In the second part we provide results about parabolic subgroups of Artin-Tits groups of spherical type: The minimal standardizer of a curve on a punctured disk is the minimal positive braid that transforms it into a round curve. We give an algorithm to compute it in a geometrical way. Then, we generalize this problem algebraically to parabolic subgroups of Artin--Tits groups of spherical type. We also show that the intersection of two parabolic subgroups is a parabolic subgroup and that the set of parabolic subgroups forms a lattice with respect to inclusion. Finally, we define the simplicial complex of irreducible parabolic subgroups, and we propose it as the analogue of the curve complex for mapping class groups

For a oriented genus g surface with one boundary component, S_g, the Torelli group is the group of orientation preserving homeomorphisms of S_g that induce the identity on homology. The Magnus representation of the Torelli group represents the action on F/F'' where F=π_1(S_g) and F'' is the second term of the derived series. I show that the kernel of the Magnus representation, Mag(S_g), is highly non-trivial and has a rich structure as a group. Specifically, I define an infinite filtration of Mag(S_g) by subgroups, called the higher order Magnus subgroups, M_k(S_g). I develop methods for generating nontrivial mapping classes in M_k(S_g) for all k and g≥2. I show that for each k the quotient M_k(S_g)/M_{k+1}(S_g) contains a subgroup isomorphic to a lower central series quotient of free groups E(g-1)_k/E(g-1)_{k+1}. Finally I show that for g≥3 the quotient M_k(S_g)/M_{k+1}(S_g) surjects onto an infinite rank torsion free abelian group. To do this, I define a Johnson-type homomorphism on each higher order Magnus subgroup quotient and show it has a highly non-trivial image.

Based on a lecture at PCMI this chapter is structured around two sets of results, one concerning groups of automorphisms of surfaces and the other concerning the nilpotent genus of groups. The first set of results exemplifies the theme that even the nicest of groups can harbour a diverse array of complicated finitely presented subgroups: we shall see that the finitely presented subgroups of the mapping class groups of surfaces of finite type can be much wilder than had been previously recognised. The second set of results fits into the quest to understand which properties of a finitely generated group can be detected by examining the group’s finite and nilpotent quotients and which cannot.

This chapter deals with finite subgroups of the mapping class group. It first explains the distinction between finite-order mapping classes and finite-order homeomorphisms, focusing on the Nielsen realization theorem for cyclic groups and detection of torsion with the symplectic representation. It then considers the problem of finding an Euler characteristic for orbifolds, to prove a Gauss–Bonnet theorem for orbifolds, and to use these results to show that there is a universal lower bound of π‎/21 for the area of any 2-dimensional orientable hyperbolic orbifold. The chapter demonstrates that, when g is greater than or equal to 2, finite subgroups have order at most 84(g − 1) and cyclic subgroups have order at most 4g + 2. It also describes finitely many conjugacy classes of finite subgroups in Mod(S) and concludes by proving that Mod(Sɡ) is generated by finitely many elements of order 2.