Geometriae Dedicata

, Volume 144, Issue 1, pp 171–190 | Cite as

A chord diagrammatic presentation of the mapping class group of a once bordered surface

Original Paper


The Ptolemy groupoid is a combinatorial groupoid generated by elementary moves on marked trivalent fatgraphs with three types of relations. Through the fatgraph decomposition of Teichmüller space, the Ptolemy groupoid is a mapping class group equivariant subgroupoid of the fundamental path groupoid of Teichmüller space with a discrete set objects. In particular, it leads to an infinite, but combinatorially simple, presentation of the mapping class group of an orientable surface. In this note, we give a presentation of a full mapping class group equivariant subgroupoid of the Ptolemy groupoid of an orientable surface with one boundary component in terms of marked linear chord diagrams, with chord slides as generators and five types of relations. We also introduce a dual version of this presentation which has advantages for certain applications, one of which is given.


Mapping class groups Ptolemy groupoid Fatgraphs Ribbon graphs Chord diagrams 

Mathematics Subject Classification (1991)

20F38 05C25 20F34 57M99 32G15 14H10 20F99 


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Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Department of MathematicsAarhus UniversityAarhus CDenmark
  2. 2.Department of MathematicsUniversity of Southern CaliforniaLos AngelesUSA

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