The mapping class group of the Cantor tree has only geometric normal 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.

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A note on the normal subgroups of mapping class groups

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100063957 ◽

1986 ◽

Vol 99 (1) ◽

pp. 79-87 ◽

Cited By ~ 16

Author(s):

D. D. Long

Keyword(s):

Mapping Class Group ◽

Class Group ◽

Mapping Class Groups ◽

Orientable Surface ◽

Faithful Representation ◽

Mapping Class ◽

Normal Subgroups ◽

Class Groups ◽

Closed Orientable Surface

0. If Fg is a closed, orientable surface of genus g, then the mapping class group of Fg is the group whose elements are orientation preserving self homeomorphisms of Fg modulo isotopy. We shall denote this group by Mg. Recall that a group is said to be linear if it admits a faithful representation as a group of matrices (where the entries for this purpose will be in some field).

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Generating the Mapping Class Group

10.23943/princeton/9780691147949.003.0005 ◽

2017 ◽

Author(s):

Benson Farb ◽

Dan Margalit

Keyword(s):

Exact Sequence ◽

Simplicial Complex ◽

Mapping Class Group ◽

Class Group ◽

Mapping Class ◽

Detailed Structure ◽

Inductive Step ◽

Closed Curves ◽

Combinatorial Object ◽

Generating Sets

This chapter considers the Dehn–Lickorish theorem, which states that when g is greater than or equal to 0, the mapping class group Mod(Sɡ) is generated by finitely many Dehn twists about nonseparating simple closed curves. The theorem is proved by induction on genus, and the Birman exact sequence is introduced as the key step for the induction. The key to the inductive step is to prove that the complex of curves C(Sɡ) is connected when g is greater than or equal to 2. The simplicial complex C(Sɡ) is a useful combinatorial object that encodes intersection patterns of simple closed curves in Sɡ. More detailed structure of C(Sɡ) is then used to find various explicit generating sets for Mod(Sɡ), including those due to Lickorish and to Humphries.

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The local-to-global property for Morse quasi-geodesics

Mathematische Zeitschrift ◽

10.1007/s00209-021-02811-w ◽

2021 ◽

Author(s):

Jacob Russell ◽

Davide Spriano ◽

Hung Cong Tran

Keyword(s):

Mapping Class Group ◽

Class Group ◽

Type Theorem ◽

Hyperbolic Groups ◽

Global Property ◽

Mapping Class ◽

Relatively Hyperbolic Groups ◽

Convex Cocompact ◽

Relatively Hyperbolic ◽

Translation Length

AbstractWe show the mapping class group, $${{\,\mathrm{CAT}\,}}(0)$$ CAT ( 0 ) groups, the fundamental groups of closed 3-manifolds, and certain relatively hyperbolic groups have a local-to-global property for Morse quasi-geodesics. This allows us to generalize combination theorems of Gitik for quasiconvex subgroups of hyperbolic groups to the stable subgroups of these groups. In the case of the mapping class group, this gives combination theorems for convex cocompact subgroups. We show a number of additional consequences of this local-to-global property, including a Cartan–Hadamard type theorem for detecting hyperbolicity locally and discreteness of translation length of conjugacy classes of Morse elements with a fixed gauge. To prove the relatively hyperbolic case, we develop a theory of deep points for local quasi-geodesics in relatively hyperbolic spaces, extending work of Hruska.

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Finiteness for mapping class group representations from twisted Dijkgraaf–Witten theory

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216518500438 ◽

2018 ◽

Vol 27 (06) ◽

pp. 1850043 ◽

Cited By ~ 1

Author(s):

Paul P. Gustafson

Keyword(s):

Braid Group ◽

Mapping Class Group ◽

Class Group ◽

Braid Groups ◽

Fusion Category ◽

Mapping Class ◽

Group Representations ◽

Colored Graphs ◽

Finite Image ◽

Spanning Set

We show that any twisted Dijkgraaf–Witten representation of a mapping class group of an orientable, compact surface with boundary has finite image. This generalizes work of Etingof et al. showing that the braid group images are finite [P. Etingof, E. C. Rowell and S. Witherspoon, Braid group representations from twisted quantum doubles of finite groups, Pacific J. Math. 234 (2008)(1) 33–42]. In particular, our result answers their question regarding finiteness of images of arbitrary mapping class group representations in the affirmative. Our approach is to translate the problem into manipulation of colored graphs embedded in the given surface. To do this translation, we use the fact that any twisted Dijkgraaf–Witten representation associated to a finite group [Formula: see text] and 3-cocycle [Formula: see text] is isomorphic to a Turaev–Viro–Barrett–Westbury (TVBW) representation associated to the spherical fusion category [Formula: see text] of twisted [Formula: see text]-graded vector spaces. The representation space for this TVBW representation is canonically isomorphic to a vector space of [Formula: see text]-colored graphs embedded in the surface [A. Kirillov, String-net model of Turaev-Viro invariants, Preprint (2011), arXiv:1106.6033 ]. By analyzing the action of the Birman generators [J. Birman, Mapping class groups and their relationship to braid groups, Comm. Pure Appl. Math. 22 (1969) 213–242] on a finite spanning set of colored graphs, we find that the mapping class group acts by permutations on a slightly larger finite spanning set. This implies that the representation has finite image.

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AUTOMORPHISMS OF BRAID GROUPS ON S2, T2, P2 AND THE KLEIN BOTTLE K

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216508005951 ◽

2008 ◽

Vol 17 (01) ◽

pp. 47-53 ◽

Cited By ~ 2

Author(s):

PING ZHANG

Keyword(s):

Braid Group ◽

Mapping Class Group ◽

Class Group ◽

Klein Bottle ◽

Closed Surface ◽

Group Extension ◽

Braid Groups ◽

Mapping Class ◽

Group B ◽

Central Automorphisms

It is shown that for the braid group Bn(M) on a closed surface M of nonnegative Euler characteristic, Out (Bn(M)) is isomorphic to a group extension of the group of central automorphisms of Bn(M) by the extended mapping class group of M, with an explicit and complete description of Aut (Bn(S2)), Aut (Bn(P2)), Out (Bn(S2)) and Out (Bn(P2)).

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ON POWERS OF HALF-TWISTS IN M(0, 2n)

Glasgow Mathematical Journal ◽

10.1017/s0017089517000143 ◽

2017 ◽

Vol 60 (2) ◽

pp. 333-338 ◽

Cited By ~ 1

Author(s):

GREGOR MASBAUM

Keyword(s):

Mapping Class Group ◽

Class Group ◽

Normal Closure ◽

Mapping Class ◽

Infinite Index ◽

Skein Theory ◽

Half Twist

AbstractWe use elementary skein theory to prove a version of a result of Stylianakis (Stylianakis, The normal closure of a power of a half-twist has infinite index in the mapping class group of a punctured sphere, arXiv:1511.02912) who showed that under mild restrictions on m and n, the normal closure of the mth power of a half-twist has infinite index in the mapping class group of a sphere with 2n punctures.

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