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$SL_n$-character varieties as spaces of graphs

Author(s): Adam S. Sikora
Journal: Trans. Amer. Math. Soc. 353 (2001), 2773 - 2804.
MSC (1991): Primary 20C15, 57M27
Posted: March 15, 2001
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Abstract:

An $SL_n$-character of a group $G$ is the trace of an $SL_n$-representation of $G.$ We show that all algebraic relations between $SL_n$-characters of $G$ can be visualized as relations between graphs (resembling Feynman diagrams) in any topological space $X,$ with $\pi_1(X)=G.$ We also show that all such relations are implied by a single local relation between graphs. In this way, we provide a topological approach to the study of $SL_n$-representations of groups.

The motivation for this paper was our work with J. Przytycki on invariants of links in 3-manifolds which are based on the Kauffman bracket skein relation. These invariants lead to a notion of a skein module of $M$ which, by a theorem of Bullock, Przytycki, and the author, is a deformation of the $SL_2$-character variety of $\pi_1(M).$This paper provides a generalization of this result to all $SL_n$-character varieties.

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Additional Information:

Adam S. Sikora
Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742
Email: asikora@math.umd.edu

DOI: 10.1090/S0002-9947-01-02700-3
PII: S 0002-9947(01)02700-3
Keywords: Character, character variety, skein module
Received by editor(s): May 18, 1999
Received by editor(s) in revised form: August 9, 2000
Posted: March 15, 2001
Additional Notes: Partially supported by NSF grant DMS93-22675
Copyright of article: Copyright 2001, American Mathematical Society

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