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Let M be a two cusped hyperbolic 3-manifold and let M(r) be the result of r Dehn filling of a fixed cusp of M. We study canonical components of the SL(2,C) character varieties of M(r). We show that the gonality of these sets is bounded, independent of the filling parameter. We also obtain bounds, depending on r, for the genus of these sets. We compute the gonality for the double twist knots, demonstrating canonical components with arbitrarily large gonality.
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This paper is devoted to the classification of connected components of Prym eigenform loci in the strata H(2,2)^odd and H(1,1,2) in the Abelian differentials bundle in genus 3. These loci, discovered by McMullen are GL^+(2,R)-invariant submanifolds (of complex dimension 3) that project to the locus of Riemann surfaces whose Jacobian variety has a factor admitting real multiplication by some quadratic order Ord_D. It turns out that these subvarieties can be classified by the discriminant D of the corresponding quadratic orders. However there algebraic varieties are not necessarily irreducible. The main result we show is that for each discriminant D the corresponding locus has one component if D is congruent to 0 or 4 mod 8, two components if D is congruent to 1 mod 8, and is empty otherwise. Our result contrasts with the case of Prym eigenform loci in the strata H(1,1) (studied by McMullen) that is connected for every discriminant D.
In this paper we describe the space of maximal components of the character variety of surface group representations into PSp(4,R) and Sp(4,R). For every rank 2 real Lie group of Hermitian type, we construct a mapping class group invariant complex structure on the maximal components. For the groups PSp(4,R) and Sp(4,R), we give a mapping class group invariant parameterization of each maximal component as an explicit holomorphic fiber bundle over Teichmuller space. Special attention is put on the connected components which are singular, we give a precise local description of the singularities and their geometric interpretation. We also describe the quotient of the maximal components for PSp(4,R) and Sp(4,R) by the action of the mapping class group as a holomorphic submersion over the moduli space of curves. These results are proven in two steps, first we use Higgs bundles to give a non-mapping class group equivariant parameterization, then we prove an analogue of Labouries conjecture for maximal PSp(4,R) representations.
In arXiv:0810.2076 we presented a conjecture generalizing the Cauchy formula for Macdonald polynomials. This conjecture encodes the mixed Hodge polynomials of the representation varieties of Riemann surfaces with semi-simple conjugacy classes at the punctures. We proved several results which support this conjecture. Here we announce new results which are consequences of those of arXiv:0810.2076.
In 1983 Culler and Shalen established a way to construct essential surfaces in a 3-manifold from ideal points of the $SL_2$-character variety associated to the 3-manifold group. We present in this article an analogous construction of certain kinds of branched surfaces (which we call essential tribranched surfaces) from ideal points of the $SL_n$-character variety for a natural number $n$ greater than or equal to 3. Further we verify that such a branched surface induces a nontrivial presentation of the 3-manifold group in terms of the fundamental group of a certain 2-dimensional complex of groups.
We study the twisted Alexander polynomial from the viewpoint of the SL(2,C)-character variety of nonabelian representations of a knot group. It is known that if a knot is fibered, then the twisted Alexander polynomials associated with nonabelian SL(2,C)-representations are all monic. In this paper, we show that the converse holds for 2-bridge knots. Furthermore we show that for a 2-bridge knot there exists a curve component in the SL(2,C)-character variety such that if the knot is not fibered then there are only finitely many characters in the component for which the associated twisted Alexander polynomials are monic. We also show that for a 2-bridge knot of genus g, in the above curve component for all but finitely many characters the associated twisted Alexander polynomials have degree 4g-2.
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