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Algebraic Mapping-Class Groups of Orientable Surfaces with Boundaries

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Infinite Groups: Geometric, Combinatorial and Dynamical Aspects

Part of the book series: Progress in Mathematics ((PM,volume 248))

Abstract

Let Sg,b,p denote a surface which is connected, orientable, has genus g, has b boundary components, and has p punctures. Let Σg,b,p denote the fundamental group of Sg,b,p.

We define the algebraic mapping-class group of Sg,b,p, denoted by Outg,b,p, and observe that topologists have shown that Outg,b,p is naturally isomorphic to the topological mapping-class group of Sg,b,p.

We study the algebraic version

$$ 1 \to \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\Sigma } _{g,b,p} \to Out_{g,b \bot 1,p} \to Out_{g,b,p} \to 1 $$

of Mess’s exact sequence that arises from filling in the interior of the (b + 1)st boundary component of Sg,b+1,p.

Here Outg,b⊥1,p is the subgroup of index b + 1 in Outg,b+1,p that fixes the (b + 1)st boundary component.

If (g, b, p) is (0, 0, 0) or (0, 0, 1), then \( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\Sigma } _{g,b,p} \) is trivial. If (g, b, p) is (0, 0, 2) or (1, 0, 0), then \( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\Sigma } _{g,b,p} \) is infinite cyclic. In all other cases, \( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\Sigma } _{g,b,p} \) is the fundamental group of the unit-tangent bundle of a suitably metrized Sg,b,p, and, hence, \( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\Sigma } _{g,b,p} \) is an extension of an infinite cyclic, central subgroup by Σg,b,p.

We give a description of the conjugation action of Outg,b⊥1,p on \( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\Sigma } _{g,b,p} \) in terms of the following three ingredients: an easily-defined action of Outg,b⊥1,p on Σg,b+1,p; the natural homomorphism Σg,b+1,p → \( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\Sigma } _{g,b,p} \) ; and, a twisting-number map Σg,b+1,p → ℤ that we define.

The work of many authors has produced aesthetic presentations of the orientation-preserving mapping-class groups Out + g,b,p with b+p ≤ 1, using the DLH generators. Within the program of giving algebraic proofs to algebraic results, we apply our machinery to give an algebraic proof of a relatively small part of this work, namely that the kernel \( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\Sigma } _{g,0,0} \) of the map

$$ Out_{g,1,0}^ + \to Out_{g,0,0} $$

is the normal closure in Out +g,1,0 of Matsumoto’s A-D word (in the DLH generators).

From the algebraic viewpoint, Outg,1,0 is the group of those automorphisms of a rank-2g free group which fix or invert a given genus g surface relator, Outg,0,0 is the group of outer automorphisms of the genus g surface group, and \( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\Sigma } _{g,0,0} \) is the kernel of the natural map between these groups. What we study are presentations for \( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\Sigma } _{g,0,0} \) , both as a group and as an Outg,1,0-group, and related topics.

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Author information

Authors and Affiliations

  1. Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193, Bellaterra (Barcelona), Spain

    Warren Dicks

  2. Department of Mathematics, The Pennsylvania State University, University Park, PA, 16802, USA

    Edward Formanek

Authors
  1. Warren Dicks
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  2. Edward Formanek
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Editor information

Editors and Affiliations

  1. Institut de Mathématiques, EPFL, 1015, Lausanne, Switzerland

    Laurent Bartholdi

  2. Dipartimento di Ingegneria, Università del Sannio, C.so Garibaldi 108, 82100, Benevento, Italy

    Tullio Ceccherini-Silberstein

  3. Section de mathématiques, Université de Genève, 1211, Geneva 4, Switzerland

    Tatiana Smirnova-Nagnibeda

  4. Institut de Mathématiques, Université Paris 7, 175 rue du Chevaleret, 75013, Paris, France

    Andrzej Zuk

Additional information

To Slava Grigorchuk on the occasion of his 50th birthday.

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© 2005 Birkhäuser Verlag Basel/Switzerland

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Dicks, W., Formanek, E. (2005). Algebraic Mapping-Class Groups of Orientable Surfaces with Boundaries. In: Bartholdi, L., Ceccherini-Silberstein, T., Smirnova-Nagnibeda, T., Zuk, A. (eds) Infinite Groups: Geometric, Combinatorial and Dynamical Aspects. Progress in Mathematics, vol 248. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7447-0_4

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