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Dimension of character varieties for $ 3$-manifolds


Authors: E. Falbel and A. Guilloux
Journal: Proc. Amer. Math. Soc. 145 (2017), 2727-2737
MSC (2010): Primary 57M27
DOI: https://doi.org/10.1090/proc/13394
Published electronically: February 6, 2017
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Abstract: Let $ M$ be an orientable $ 3$-manifold, compact with boundary and $ \Gamma $ its fundamental group. Consider a complex reductive algebraic group $ G$. The character variety $ X(\Gamma ,G)$ is the GIT quotient $ \textup {Hom}(\Gamma ,G)//G$ of the space of morphisms $ \Gamma \to G$ by the natural action by conjugation of $ G$. In the case $ G=\textup {SL}(2,\mathbb{C})$ this space has been thoroughly studied.

Following work of Thurston (1980), as presented by Culler-Shalen (1983), we give a lower bound for the dimension of irreducible components of $ X(\Gamma ,G)$ in terms of the Euler characteristic $ \chi (M)$ of $ M$, the number $ t$ of torus boundary components of $ M$, the dimension $ d$ and the rank $ r$ of $ G$. Indeed, under mild assumptions on an irreducible component $ X_0$ of $ X(\Gamma ,G)$, we prove the inequality

$\displaystyle \mathrm {dim}(X_0)\geq t \cdot r - d\chi (M).$


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Additional Information

E. Falbel
Affiliation: Institut de Mathématiques de Jussieu-Paris Rive Gauche, Unité Mixte de Recherche 7586 du CNRS, CNRS UMR 7586
Email: elisha.falbel@imj-prg.fr

A. Guilloux
Affiliation: INRIA EPI-OURAGAN, Université Pierre et Marie Curie, 4 place Jussieu 75252 Paris Cédex, France - and - Institut de Mathématiques de Jussieu-Paris Rive Gauche, Université Pierre et Marie Curie, 4 Place Jussieu, 75252 Paris Cédex, France
Email: antonin.guilloux@imj-prg.fr

DOI: https://doi.org/10.1090/proc/13394
Received by editor(s): December 15, 2015
Received by editor(s) in revised form: April 6, 2016
Published electronically: February 6, 2017
Additional Notes: This work was supported in part by the ANR through the project “Structures Géométriques et Triangulations”.
Communicated by: Michael Wolf
Article copyright: © Copyright 2017 American Mathematical Society