Groupoid extensions of mapping class representations for bordered surfaces
type: Item
instance of: scholarly article
title: Groupoid extensions of mapping class representations for bordered surfaces
zbMATH Open document ID: 1196.57013
author: J. E. Andersen
author: Alex James Bene
author: Robert Penner
published in: Topology and its Applications
publication date: 2009-11-05T00:00:00Z
full work available at URL: https://arxiv.org/abs/0710.2651
review text: Let us first quote the authors' abstract: ``The mapping class group of a surface with one boundary component admits numerous interesting representations including a representation as a group of automorphisms of a free group and as a group of symplectic transformations. Insofar as the mapping class group can be identified with the fundamental group of Riemann's moduli space, it is furthermore identified with a subgroup of the fundamental path groupoid upon choosing a basepoint. A combinatorial model for this, the mapping class groupoid, arises from the invariant cell decomposition of Teichmüller space, whose fundamental path groupoid is called the Ptolemy groupoid. It is natural to try to extend representations of the mapping class group to the mapping class groupoid, i.e., to construct a homomorphism from the mapping class groupoid to the same target that extends the given representations arising from various choices of basepoint. Among others, we extend both aforementioned representations to the groupoid level in this sense, where the symplectic representation is lifted both rationally and integrally. The techniques of proof include several algorithms involving fatgraphs and chord diagrams. The former extension is given by explicit formulae depending upon six essential cases, and the kernel and image of the groupoid representation are computed. Furthermore, this provides groupoid extensions of any representation of the mapping class group that factors through its action on the fundamental group of the surface including, for instance, the Magnus representation and representation on the moduli spaces of flat connections.'' Let us describe the content of the paper in more detail. Section~1 is an introduction. In Section~2 the notion of fatgraph is reminded. Fatgraph is a graph with a cyclic ordering of oriented edges pointing to every fixed vertex. Boundary cycles of a fatgraph and its genus are defined naturally. A (once-)bordered fatgraph is a fatgraph with a unique…
reviewed by: Samyon R. Nasyrov
Mathematics Subject Classification ID: 57M99
Mathematics Subject Classification ID: 20F38
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 32G15
Mathematics Subject Classification ID: 14H10
Mathematics Subject Classification ID: 20F99
zbMATH DE Number: 5625471
zbMATH Keywords: mapping class groups
zbMATH Keywords: Ptolemy groupoid
zbMATH Keywords: automorphisms of free groups
zbMATH Keywords: Magnus representation
zbMATH Keywords: fatgraphs
zbMATH Keywords: ribbon graphs
zbMATH Keywords: chord diagrams
zbMATH Keywords: bordered surface
zbMATH Keywords: fatgraph
zbMATH Keywords: Whitehead move
MaRDI profile type: Publication
OpenAlex ID: W2051264796
arXiv ID: 0710.2651
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DOI: 10.1016/J.TOPOL.2009.08.001
description: scientific article