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Journal articles on the topic 'Mapping class subgroups'

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Published: 9 November 2024

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1

Matsuzaki, Katsuhiko. "Polycyclic quasiconformal mapping class subgroups." Pacific Journal of Mathematics 251, no. 2 (2011): 361–74. http://dx.doi.org/10.2140/pjm.2011.251.361.

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2

Clay, Matt, Johanna Mangahas, and Dan Margalit. "Right-angled Artin groups as normal subgroups of mapping class groups." Compositio Mathematica 157, no. 8 (2021): 1807–52. http://dx.doi.org/10.1112/s0010437x21007417.

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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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3

Calegari, Danny, and Lvzhou Chen. "Normal subgroups of big mapping class groups." Transactions of the American Mathematical Society, Series B 9, no. 30 (2022): 957–76. http://dx.doi.org/10.1090/btran/108.

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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.
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4

Kim, Heejoung. "Stable subgroups and Morse subgroups in mapping class groups." International Journal of Algebra and Computation 29, no. 05 (2019): 893–903. http://dx.doi.org/10.1142/s0218196719500346.

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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.
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5

Leininger, Christopher J., and D. B. McReynolds. "Separable subgroups of mapping class groups." Topology and its Applications 154, no. 1 (2007): 1–10. http://dx.doi.org/10.1016/j.topol.2006.03.013.

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6

Bavard, Juliette, Spencer Dowdall, and Kasra Rafi. "Isomorphisms Between Big Mapping Class Groups." International Mathematics Research Notices 2020, no. 10 (2018): 3084–99. http://dx.doi.org/10.1093/imrn/rny093.

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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.
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7

Farb, Benson, and Lee Mosher. "Convex cocompact subgroups of mapping class groups." Geometry & Topology 6, no. 1 (2002): 91–152. http://dx.doi.org/10.2140/gt.2002.6.91.

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8

Berrick, A. J., V. Gebhardt, and L. Paris. "Finite index subgroups of mapping class groups." Proceedings of the London Mathematical Society 108, no. 3 (2013): 575–99. http://dx.doi.org/10.1112/plms/pdt022.

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9

Anderson, James W., Javier Aramayona, and Kenneth J. Shackleton. "Free subgroups of surface mapping class groups." Conformal Geometry and Dynamics of the American Mathematical Society 11, no. 04 (2007): 44–55. http://dx.doi.org/10.1090/s1088-4173-07-00156-7.

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10

Franks, John, and Kamlesh Parwani. "Zero entropy subgroups of mapping class groups." Geometriae Dedicata 186, no. 1 (2016): 27–38. http://dx.doi.org/10.1007/s10711-016-0178-9.

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11

Szepietowski, Błażej. "On finite index subgroups of the mapping class group of a nonorientable surface." Glasnik Matematicki 49, no. 2 (2014): 337–50. http://dx.doi.org/10.3336/gm.49.2.08.

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12

KENT IV, RICHARD P., and CHRISTOPHER J. LEININGER. "Uniform convergence in the mapping class group." Ergodic Theory and Dynamical Systems 28, no. 4 (2008): 1177–95. http://dx.doi.org/10.1017/s0143385707000818.

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AbstractWe characterize convex cocompact subgroups of the mapping class group of a surface in terms of uniform convergence actions on the zero locus of the limit set. We also construct subgroups that act as uniform convergence groups on their limit sets, but are not convex cocompact.
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13

Bestvina, Mladen, and Koji Fujiwara. "Bounded cohomology of subgroups of mapping class groups." Geometry & Topology 6, no. 1 (2002): 69–89. http://dx.doi.org/10.2140/gt.2002.6.69.

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14

Broughton, Allen, and Aaron Wootton. "Finite abelian subgroups of the mapping class group." Algebraic & Geometric Topology 7, no. 4 (2007): 1651–97. http://dx.doi.org/10.2140/agt.2007.7.1651.

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15

Masbaum, G., and A. W. Reid. "Frattini and related subgroups of mapping class groups." Proceedings of the Steklov Institute of Mathematics 292, no. 1 (2016): 143–52. http://dx.doi.org/10.1134/s0081543816010090.

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16

Masbaum, G., and A. W. Reid. "Frattini and Related Subgroups of Mapping Class Groups." Труды математического института им. Стеклова 292, no. 01 (2016): 149–58. http://dx.doi.org/10.1134/s037196851601009x.

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17

Stukow, Michal. "Commensurability of geometric subgroups of mapping class groups." Geometriae Dedicata 143, no. 1 (2009): 117–42. http://dx.doi.org/10.1007/s10711-009-9377-y.

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18

Whittlesey, Kim. "Normal all pseudo-Anosov subgroups of mapping class groups." Geometry & Topology 4, no. 1 (2000): 293–307. http://dx.doi.org/10.2140/gt.2000.4.293.

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19

Berrick, A. J., E. Hanbury, and J. Wu. "Brunnian subgroups of mapping class groups and braid groups." Proceedings of the London Mathematical Society 107, no. 4 (2013): 875–906. http://dx.doi.org/10.1112/plms/pds096.

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20

Liang, Hao. "Centralizers of finite subgroups of the mapping class group." Algebraic & Geometric Topology 13, no. 3 (2013): 1513–30. http://dx.doi.org/10.2140/agt.2013.13.1513.

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21

Vlamis, Nicholas. "Quasiconformal homogeneity and subgroups of the mapping class group." Michigan Mathematical Journal 64, no. 1 (2015): 53–75. http://dx.doi.org/10.1307/mmj/1427203285.

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22

Aramayona, Javier, and Louis Funar. "Quotients of the mapping class group by power subgroups." Bulletin of the London Mathematical Society 51, no. 3 (2019): 385–98. http://dx.doi.org/10.1112/blms.12236.

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23

Anderson, James W., Javier Aramayona, and Kenneth J. Shackleton. "Corrigendum to ‘‘Free subgroups of surface mapping class groups”." Conformal Geometry and Dynamics of the American Mathematical Society 13, no. 07 (2009): 136–38. http://dx.doi.org/10.1090/s1088-4173-09-00193-3.

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24

Leininger, Christopher, and Jacob Russell. "Pseudo-Anosov subgroups of general fibered 3–manifold groups." Transactions of the American Mathematical Society, Series B 10, no. 32 (2023): 1141–72. http://dx.doi.org/10.1090/btran/157.

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We show that finitely generated and purely pseudo-Anosov subgroups of fundamental groups of fibered 3–manifolds with reducible monodromy are convex cocompact as subgroups of the mapping class group via the Birman exact sequence. Combined with results of Dowdall–Kent–Leininger and Kent–Leininger–Schleimer, this establishes the result for the image of all such fibered 3–manifold groups in the mapping class group.
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25

AKITA, TOSHIYUKI, and NARIYA KAWAZUMI. "Integral Riemann–Roch formulae for cyclic subgroups of mapping class groups." Mathematical Proceedings of the Cambridge Philosophical Society 144, no. 2 (2008): 411–21. http://dx.doi.org/10.1017/s0305004107001016.

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AbstractThe first author conjectured certain relations for Morita–Mumford classes and Newton classes in the integral cohomology of mapping class groups (integral Riemann–Roch formulae). In this paper, the conjecture is verified for cyclic subgroups of mapping class groups.
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26

McCarthy, John. "A "Tits-Alternative" for Subgroups of Surface Mapping Class Groups." Transactions of the American Mathematical Society 291, no. 2 (1985): 583. http://dx.doi.org/10.2307/2000100.

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27

Keating, Ailsa M. "Dehn twists and free subgroups of symplectic mapping class groups." Journal of Topology 7, no. 2 (2013): 436–74. http://dx.doi.org/10.1112/jtopol/jtt033.

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28

McCarthy, John. "A ‘‘Tits-alternative” for subgroups of surface mapping class groups." Transactions of the American Mathematical Society 291, no. 2 (1985): 583. http://dx.doi.org/10.1090/s0002-9947-1985-0800253-8.

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29

Long, D. D. "A note on the normal subgroups of mapping class groups." Mathematical Proceedings of the Cambridge Philosophical Society 99, no. 1 (1986): 79–87. http://dx.doi.org/10.1017/s0305004100063957.

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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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30

Behrstock, Jason, and Dan Margalit. "Curve Complexes and Finite Index Subgroups of Mapping Class Groups." Geometriae Dedicata 118, no. 1 (2006): 71–85. http://dx.doi.org/10.1007/s10711-005-9022-3.

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31

OHSHIKA, Ken'ichi. "Finite subgroups of mapping class groups of geometric $3$ -manifolds." Journal of the Mathematical Society of Japan 39, no. 3 (1987): 447–54. http://dx.doi.org/10.2969/jmsj/03930447.

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32

Dicks, Warren, and Edward Formanek. "Automorphism Subgroups of Finite Index in Algebraic Mapping Class Groups." Journal of Algebra 189, no. 1 (1997): 58–89. http://dx.doi.org/10.1006/jabr.1996.6876.

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33

Brendle, Tara E., and Dan Margalit. "Normal subgroups of mapping class groups and the metaconjecture of Ivanov." Journal of the American Mathematical Society 32, no. 4 (2019): 1009–70. http://dx.doi.org/10.1090/jams/927.

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34

Humphries, Stephen P. "An action of subgroups of mapping class groups on polynomial algebras." Topology and its Applications 154, no. 6 (2007): 1053–83. http://dx.doi.org/10.1016/j.topol.2006.10.009.

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35

Leininger, C. J., and A. W. Reid. "A combination theorem for Veech subgroups of the mapping class group." GAFA Geometric And Functional Analysis 16, no. 2 (2006): 403–36. http://dx.doi.org/10.1007/s00039-006-0556-9.

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36

Mangahas, Johanna. "Uniform Uniform Exponential Growth of Subgroups of the Mapping Class Group." Geometric and Functional Analysis 19, no. 5 (2009): 1468–80. http://dx.doi.org/10.1007/s00039-009-0038-y.

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37

GARDINER, F. P., and N. LAKIC. "A VECTOR FIELD APPROACH TO MAPPING CLASS ACTIONS." Proceedings of the London Mathematical Society 92, no. 2 (2006): 403–27. http://dx.doi.org/10.1112/s0024611505015558.

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We present a vector field method for showing that certain subgroups of the mapping class group $\Gamma$ of a Riemann surface of infinite topological type act properly discontinuously. We apply the method to the group of homotopy classes of quasiconformal self-maps of the complement $\Omega$ of a Cantor set in $\mathbb{C}$. When the Cantor set has bounded geometric type, we show that $\Gamma(\Omega)$ acts on the Teichmüller space $T(\Omega)$ properly discontinuously. Also, we apply the same method to show that the pure mapping class group $\Gamma_0(\Omega \cup \{\infty\})$ acts properly discontinuously on $T(\Omega \cup \{\infty\})$.
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38

Ohshika, Ken’ichi, and Makoto Sakuma. "Subgroups of mapping class groups related to Heegaard splittings and bridge decompositions." Geometriae Dedicata 180, no. 1 (2015): 117–34. http://dx.doi.org/10.1007/s10711-015-0094-4.

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39

Lanier, Justin, and Marissa Loving. "Centers of subgroups of big mapping class groups and the Tits alternative." Glasnik Matematicki 55, no. 1 (2020): 85–91. http://dx.doi.org/10.3336/gm.55.1.07.

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40

Bridson, Martin R. "On the subgroups of right-angled Artin groups and mapping class groups." Mathematical Research Letters 20, no. 2 (2013): 203–12. http://dx.doi.org/10.4310/mrl.2013.v20.n2.a1.

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41

Hadari, Asaf. "Non virtually solvable subgroups of mapping class groups have non virtually solvable representations." Groups, Geometry, and Dynamics 14, no. 4 (2020): 1333–50. http://dx.doi.org/10.4171/ggd/583.

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42

Putman, Andrew, and Ben Wieland. "Abelian quotients of subgroups of the mapping class group and higher Prym representations." Journal of the London Mathematical Society 88, no. 1 (2013): 79–96. http://dx.doi.org/10.1112/jlms/jdt001.

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43

McCarthy, John D. "On the first cohomology group of cofinite subgroups in surface mapping class groups." Topology 40, no. 2 (2001): 401–18. http://dx.doi.org/10.1016/s0040-9383(99)00066-x.

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44

KAPOVICH, ILYA, and MARTIN LUSTIG. "PING-PONG AND OUTER SPACE." Journal of Topology and Analysis 02, no. 02 (2010): 173–201. http://dx.doi.org/10.1142/s1793525310000318.

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We prove that, if φ, ψ ∈ Out (FN) are hyperbolic iwips (irreducible with irreducible powers) such that 〈φ, ψ〉 ⊆ Out (FN) is not virtually cyclic, then some high powers of φ and ψ generate a free subgroup of rank two for which all nontrivial elements are again hyperbolic iwips. Being a hyperbolic iwip element of Out (FN) is strongly analogous to being a pseudo-Anosov element of a mapping class group, so the above result provides analogues of "purely pseudo-Anosov" free subgroups in Out (FN).
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45

Putman, Andrew. "The Johnson homomorphism and its kernel." Journal für die reine und angewandte Mathematik (Crelles Journal) 2018, no. 735 (2018): 109–41. http://dx.doi.org/10.1515/crelle-2015-0017.

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AbstractWe give a new proof of a celebrated theorem of Dennis Johnson that asserts that the kernel of the Johnson homomorphism on the Torelli subgroup of the mapping class group is generated by separating twists. In fact, we prove a more general result that also applies to “subsurface Torelli groups”. Using this, we extend Johnson’s calculation of the rational abelianization of the Torelli group not only to the subsurface Torelli groups, but also to finite-index subgroups of the Torelli group that contain the kernel of the Johnson homomorphism.
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46

Lee, Chun-Nip. "Farrell cohomology and centralizets of elementary abelian p-subgroups." Mathematical Proceedings of the Cambridge Philosophical Society 119, no. 3 (1996): 403–17. http://dx.doi.org/10.1017/s0305004100074302.

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Let Γ be a discrete group. Γ is said to have finite virtual cohomological dimension (vcd) if there exists a finite index torsion-free subgroup Γ′ of G such that Γ′ has finite cohomological dimension over ℤ. Examples of such groups include finite groups, fundamental group of a finite graph of finite groups, arithmetic groups, mapping class groups and outer automorphism groups of free groups. One of the fundamental problems in topology is to understand the cohomology of these finite vcd-groups.
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47

Juan-Pineda, Daniel, and Alejandra Trujillo-Negrete. "On classifying spaces for the family of virtually cyclic subgroups in mapping class groups." Pure and Applied Mathematics Quarterly 12, no. 2 (2016): 261–92. http://dx.doi.org/10.4310/pamq.2016.v12.n2.a4.

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48

Putman, Andrew. "A note on the abelianizations of finite-index subgroups of the mapping class group." Proceedings of the American Mathematical Society 138, no. 02 (2009): 753–58. http://dx.doi.org/10.1090/s0002-9939-09-10124-7.

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49

Kitano, Teruaki. "Johnson's homomorphisms of subgroups of the mapping class group, the Magnus expansion and Massey higher products of mapping tori." Topology and its Applications 69, no. 2 (1996): 165–72. http://dx.doi.org/10.1016/0166-8641(95)00077-1.

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50

UEMURA, Takeshi. "Morita-Mumford classes on finite cyclic subgroups of the mapping class group of closed surfaces." Hokkaido Mathematical Journal 28, no. 3 (1999): 597–611. http://dx.doi.org/10.14492/hokmj/1351001239.

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