Restrictions of representations of a surface group to a pair of free subgroups
type: Item
instance of: scholarly article
title: Restrictions of representations of a surface group to a pair of free subgroups
zbMATH Open document ID: 0964.57001
author: Benjamin Martin
published in: Journal of Algebra
publication date: 2000-08-13T00:00:00Z
review text: The fundamental group \(\Pi\) of a compact orientable genus \(m\) surface has a standard presentation of the form \[ \Pi=\bigl \langle a_1,b_1, \dots,a_m, b_m\mid [a_1,b_1] \cdots [a_m,b_m]= 1\bigr\rangle. \] In the paper under review the author studies representations of the group \(\Pi\) and its special free subgroups \(A=\langle a_1,\dots, a_m\rangle\) and \(B=\langle b_1, \dots, b_m\rangle\) of rank \(m\) in a connected reductive algebraic group \(G\) over an algebraically closed field \(k\) of characteristic zero. Let \(C(F,G)\) denote the variety of closed conjugacy classes of representations of a group \(F\) in \(G\). Then the product of the restrictions of the representations associated with the two subgroups \(A\) and \(B\) of the fundamental group \(\Pi\) gives rise to an induced morphism \(r_{A,B}: C(\Pi,G)\to C(A,G) \times C(B,G)\). The main theorem of the paper states that if the genus \(m\) of the surface exceeds the semisimple rank \(s\) of \(G\), then this morphism is dominant and almost all fibres are finite; precisely, if \(m\geq s+1\) then there exists a non-empty open subset \(U\) of \(C(A,G)\times C(B,G)\) such that all fibres of the morphism \(r_{A,B}\) over the set \(U\) are finite and non-empty. From this theorem some related results concerning the associated varieties of representations are also derived. Finally, as an example, the case \(G=GL_4(k)\) is studied in detail.
reviewed by: Frank Neumann
Mathematics Subject Classification ID: 57M05
Mathematics Subject Classification ID: 14F35
Mathematics Subject Classification ID: 20F34
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 57N05
zbMATH DE Number: 1423611
zbMATH Keywords: automorphisms
zbMATH Keywords: representation theory
zbMATH Keywords: fundamental group
MaRDI profile type: Publication
full work available at URL: jabr.1999.8110
OpenAlex ID: W2006718435
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DOI: 10.1006/JABR.1999.8110
description: scientific article; zbMATH DE number 1423611