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Title:
Mapping Class Group Actions on Character Varieties
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Abstract:
In this talk we study the action of the mapping class group on the SL(2,C)-character variety. In particular, we focus on the character variety of the free group F_3 of rank 3, considered as the fundamental group of the four-holed sphere S. We define a domain of discontinuity for this action which strictly contains the interior of discrete and faithful representations. We will describe the combinatorial view point we adopt using trace functions on simple closed curves, and sketch the main ideas of the proof. In the case of real characters, we show that this domain of discontinuity may be non-empty on the components where the relative Euler class is non-maximal. Time permitting, we will also discuss the more recent work with F. Palesi where we see F_3 as the fundamental group of the three-holed projective plane N, and the various open questions on the topic.
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