𝑆𝐿_{𝑛}-character varieties as spaces of graphs

Sikora, Adam

doi:10.1090/S0002-9947-01-02700-3

$SL_n$-character varieties as spaces of graphs
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Trans. Amer. Math. Soc. 353 (2001), 2773-2804 Request permission

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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