Billiards and Teichmüller curves on Hilbert modular surfaces

By Curtis T. McMullen. 2003

326 citations recorded by the source corpus.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9934a780dc60823a1839fac02a0ac598165745a0

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Citations in the corpus

Teichmüller curves in genus two: The decagon and beyond (2005, background): “In contrast, the Teichm¨uller curves in HD are covered by graphs of transcendental functions [ Mc1 , §10], and their abundance depends on the rationality of √ D.”
Veech surfaces and complete periodicity in genus two (2004, methodology): “We note that a di(cid:11)erent classi(cid:12)cation of Veech surfaces in genus two, formulated in terms of Jacobians with real multiplication, was obtained by C. McMullen in [M1].”
Variations of Hodge structures of a Teichmüller curve (2006, methodology): “This group is related (see [McM1, Prop.”
An extraordinary origami curve (2005, methodology): “For an introduction to Teichm¨uller geodesics and Teichm¨uller curves we refer to [EG ] and [McM].”
INFINITELY GENERATED VEECH GROUPS (2004, background): “Proof. The proof of Lemma 2 of [HS2] shows that the Veech group of this cross is the so-called Hecke triangle group of index 5. Indeed, Lemma 4 of [HS2] shows that this cross is affinely equivalent to Veech’s [Vch] original nonarithmetic translation surface arising from copies of the regular pentago”
On the classification of rank-two representations of quasiprojective fundamental groups (2007, background): “The classical case which has been the most extensively studied is that of Hilbert modular varieties, corresponding to SL(2, F ) where F is a totally real field [30] [49] [51] [57] [84] [101] [127].”
Teichmuller curves, triangle groups, and Lyapunov exponents (2005, methodology): “We denote by || · || := || · ||T the associated metric on V . There is a natural SL2(R)-action on 1Mg obtained by post-composing the charts given by integrating the one-form with the R-linear map given by A 2 SL2(R) to obtain a new complex structure and new holomorphic one-form (see e.g. [ McM03 ] a”
Braid groups and Hodge theory (2013, methodology): “ρr(Bn). The locus F(Z) ⊂ S is a Kobayashi geodesic subvariety, reminiscent of the Teichm¨uller curves on Hilbert modular surfaces studied in [ Mc1 ] and [Mc2].”
Cylinder deformations in orbit closures of translation surfaces (2013, background): “The author thanks Curtis McMullen and the referee for helpful comments on an earlier draft.”
Veech groups without parabolic elements (2005, background): “Note that these results have been reproved in [ Mc1 , Mc2].”
Splitting mixed Hodge structures over affine invariant manifolds (2013, background): “McMullen has related in [McM03] Teich-m¨uller curves in genus 2 with real multiplication.”
Periodic points on Veech surfaces and the Mordell-Weil group over a Teichmüller curve (2004, background): “g denote the tautological bundle minus the zero section over Mg, the moduli space of curves of genus g. Its points consist of pairs (X 0 ,ω 0 ) of a Riemann surface plus a non-zero holomorphic one-form on X 0 .T here is a natural GL + (R)-action on ΩM ∗ (see [Ve89] or [ McM03 ]).”
Prym Varieties and Teichmüller Curves (2006, methodology): “A sampler. Reflection groups of finite index in GL(X, ω) were determined for several curves of the form V (Ln) in [ Mc1 , §9].”