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MEDNYKH’S FORMULA VIA LATTICE TOPOLOGICAL QUANTUM FIELD THEORIES
, 2008
"... Abstract. Mednykh [Me78] proved that for any finite group G and any orientable surface S, there is a formula for #Hom(π1(S), G) in terms of the Euler characteristic of S and the dimensions of the irreducible representations of G. A similar formula in the nonorientable case was proved by Frobenius an ..."
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Abstract. Mednykh [Me78] proved that for any finite group G and any orientable surface S, there is a formula for #Hom(π1(S), G) in terms of the Euler characteristic of S and the dimensions of the irreducible representations of G. A similar formula in the nonorientable case was proved by Frobenius and Schur [FS06]. Both of these proofs use character theory and an explicit presentation for π1. These results have been reproven using quantum field theory ([FQ93], [MY05], and others). Here we present a greatly simplified proof of these results which uses only elementary topology and combinatorics. The main tool is an elementary invariant of surfaces attached to a semisimple algebra called a lattice topological quantum field theory. 1.
VOLUME OF REPRESENTATION VARIETIES
, 2002
"... Abstract. We introduce the notion of volume of the representation variety of a finitely presented discrete group in a compact Lie group using the pushforward measure associated to a map defined by a presentation of the discrete group. We show that the volume thus defined is invariant under the Andr ..."
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Abstract. We introduce the notion of volume of the representation variety of a finitely presented discrete group in a compact Lie group using the pushforward measure associated to a map defined by a presentation of the discrete group. We show that the volume thus defined is invariant under the AndrewsCurtis moves of the generators and relators of the discrete group, and moreover, that it is actually independent of the choice of presentation if the difference of the number of generators and the number of relators remains the same. We then calculate the volume of the representation variety of a surface group in an arbitrary compact Lie group using the classical technique of Frobenius and Schur on finite groups. Our formulas recover the results of Witten and Liu on the symplectic volume and the Reidemeister torsion of the moduli space of flat Gconnections on a
GEOMETRY OF CHARACTER VARIETIES OF SURFACE GROUPS
, 710
"... Abstract. This article is based on a talk delivered at the RIMS–OCAMI Joint International ..."
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Abstract. This article is based on a talk delivered at the RIMS–OCAMI Joint International