One-holed Torus Research Articles

Length minima for an infinite family of filling closed curves on a one-holed torus

Length minima for an infinite family of filling closed curves on a one-holed torus

The Algebraic de Rham Cohomology of Representation Varieties

AbstractThe SL(2, ℂ)-representation varieties of punctured surfaces form natural families parameterized by monodromies at the punctures. In this paper, we compute the loci where these varieties are singular for the cases of one-holed and two-holed tori and the four-holed sphere. We then compute the de Rham cohomologies of these varieties of the one-holed torus and the four-holed sphere when the varieties are smooth via the Grothendieck theorem. Furthermore, we produce the explicit Gauß-Manin connection on the natural family of the smooth SL(2, ℂ)-representation varieties of the one-holed torus.

PROPER AFFINE DEFORMATIONS OF THE ONE-HOLED TORUS

A Margulis spacetime is a complete at Lorentzian 3-manifold M with free fundamental group. Associated to M is a noncompact complete hyperbolic surface ∑ homotopy-equivalent to M. The purpose of this paper is to classify Margulis spacetimes when ∑ is homeomorphic to a one-holed torus. We show that every such M decomposes into polyhedra bounded by crooked planes, corresponding to an ideal triangulation of ∑. This paper classifies and analyzes the structure of crooked ideal triangles, which play the same role for Margulis spacetimes as ideal triangles play for hyperbolic surfaces.

Stein fillable contact 3–manifolds and positive open books of genus one

A two-dimensional open book (S,h) determines a closed, oriented three-manifold Y(S,h) and a contact structure C(S,h) on Y(S,h). The contact structure C(S,h) is Stein fillable if h is positive, i.e. h can be written as a product of right-handed Dehn twists. Work of Wendl implies that when S has genus zero the converse statement holds, that is if C(S,h) is Stein fillable then h is positive. On the other hand, results by Wand and by Baker, Etnyre and Van Horn-Morris imply the existence of counterexamples to the converse statement with S of arbitrary genus strictly greater than one. The main purpose of this paper is to prove the converse statement under the assumption that S is a one-holed torus and Y(S,h) is a Heegaard Floer L-space.

New identities for small hyperbolic surfaces

Luo and Tan gave a new identity for hyperbolic surfaces with/without geodesic boundary in terms of dilogarithms of the lengths of simple closed geodesics on embedded three-holed spheres or one-holed tori. However, the identity was trivial for a hyperbolic one-holed torus with geodesic boundary. In this paper we adapt the argument from Luo and Tan to give an identity for hyperbolic tori with one geodesic boundary or cusp in terms of dilogarithm functions on the set of lengths of simple closed geodesics on the torus. As a corollary, we are also able to express the Luo-Tan identity as a sum over all immersed three-holed spheres $P$ which are embeddings when restricted to the interior of $P$.

INVOLUTIONS AND THE FRICKE SPACES OF SURFACES WITH BOUNDARY

The purpose of this paper is to find expressions of the Fricke spaces of some basic surfaces which are a three-holed sphere �(0,3), a one-holed torus �(1,1), and a four-holed sphere �(0,4). For this goal, we define the involutions corresponding to oriented axes of loxodromic elements and an inner product h , i which gives the information about locations of axes of loxodromic elements. The signs of traces of holonomy elements, which are calculated by lifting a representation from PSL(2,C) to SL(2,C), play a very important role in determining the discreteness of holonomy groups.

LIMITS OF THE QUANTUM SO(3) REPRESENTATIONS FOR THE ONE-HOLED TORUS

For N ≥ 2, we study a certain sequence [Formula: see text] of N-dimensional representations of the mapping class group of the one-holed torus arising from SO(3)-TQFT, and show that the conjecture of Andersen, Masbaum, and Ueno [Topological quantum field theory and the Nielsen–Thurston classification of M(0,4), Math. Proc. Cambridge Philos Soc.141 (2006) 477–488] holds for these representations. This is done by proving that, in a certain basis and up to a rescaling, the matrices of these representations converge as p tends to infinity. Moreover, the limits describe the action of SL2(ℤ) on the space of homogeneous polynomials of two variables of total degree N - 1.

Self-duality of the compactified Ruijsenaars–Schneider system from quasi-Hamiltonian reduction

The Delzant theorem of symplectic topology is used to derive the completely integrable compactified Ruijsenaars–Schneider IIIb system from a quasi-Hamiltonian reduction of the internally fused double SU(n)×SU(n). In particular, the reduced spectral functions depending respectively on the first and second SU(n) factor of the double engender two toric moment maps on the IIIb phase space CP(n−1) that play the roles of action-variables and particle-positions. A suitable central extension of the SL(2,Z) mapping class group of the torus with one boundary component is shown to act on the quasi-Hamiltonian double by automorphisms and, upon reduction, the standard generator S of the mapping class group is proved to descend to the Ruijsenaars self-duality symplectomorphism that exchanges the toric moment maps. We give also two new presentations of this duality map: one as the composition of two Delzant symplectomorphisms and the other as the composition of three Dehn twist symplectomorphisms realized by Goldman twist flows. Through the well-known relation between quasi-Hamiltonian manifolds and moduli spaces, our results rigorously establish the validity of the interpretation [going back to Gorsky and Nekrasov] of the IIIb system in terms of flat SU(n) connections on the one-holed torus.

Integral topological quantum field theory for a one-holed torus

We give new explicit formulas for the representations of the mapping class group of a genus one surface with one boundary component which arise from Integral TQFT. Our formulas allow one to compute the h-adic expansion of the TQFT-matrix associated to a mapping class in a straightforward way. Truncating the h-adic expansion gives an approximation of the representation by representations into finite groups. As a special case, we study the induced representations over finite fields and identify them up to isomorphism. The key technical ingredient of the paper are new bases of the Integral TQFT modules which are orthogonal with respect to the Hopf pairing. We construct these orthogonal bases in arbitrary genus, and briefly describe some other applications of them.

Topological Open Strings on Orbifolds

We use the remodeling approach to the B-model topological string in terms of recursion relations to study open string amplitudes at orbifold points. To this end, we clarify modular properties of the open amplitudes and rewrite them in a form that makes their transformation properties under the modular group manifest. We exemplify this procedure for the C^3/Z_3 orbifold point of local P^2, where we present results for topological string amplitudes for genus zero and up to three holes, and for the one-holed torus. These amplitudes can be understood as generating functions for either open orbifold Gromov-Witten invariants of C^3/Z_3, or correlation functions in the orbifold CFT involving insertions of both bulk and boundary operators.

Obtaining the one-holed torus from pants: Duality in anSL(3,ℂ)-character variety

The SL(3,C)-representation variety R of a free group F arises naturally by considering surface group representations for a surface with boundary. There is a SL(3,C)-action on the coordinate ring of R. The geometric points of the subring of invariants of this action is an affine variety X. The points of X parametrize isomorphism classes of completely reducible representations. The coordinate ring C[X] is a complex Poisson algebra with respect to a presentation of F imposed by the surface. In previous work, we have worked out the bracket on all generators when the surface is a three-holed sphere and when the surface is a one-holed torus. In this paper, we show how the symplectic leaves corresponding to these two different Poisson structures on X relate to each other. In particular, they are symplectically dual at a generic point. Moreover, the topological gluing map which turns the three-holed sphere into the one-holed torus induces a rank preserving Poisson map on C[X].

Poisson geometry of 𝑆𝐿(3,ℂ)-character varieties relative to a surface with boundary

TheSL(3,C)\mathrm {SL}(3,\mathbb {C})-representation varietyR\mathfrak {R}of a free groupFr\mathtt {F}_rarises naturally by considering surface group representations for a surface with boundary. There is anSL(3,C)\mathrm {SL}(3,\mathbb {C})-action on the coordinate ring ofR\mathfrak {R}by conjugation. The geometric points of the subring of invariants of this action is an affine varietyX\mathfrak {X}. The points ofX\mathfrak {X}parametrize isomorphism classes of completely reducible representations. We show the coordinate ringC[X]\mathbb {C}[\mathfrak {X}]is a complex Poisson algebra with respect to a presentation ofFr\mathtt {F}_rimposed by the surface. Lastly, we work out the bracket on all generators when the surface is a three-holed sphere or a one-holed torus.

End Invariants for SL(2,C) Characters of the One-Holed Torus

We define and study the set ε(ρ) of end invariants of an SL(2,C) character ρ of the one-holed torus T . We show that the set ε(ρ) is the entire projective lamination space [inline-graphic 01] of T if and only if ρ corresponds to the dihedral representation or ρ is real and corresponds to an SU(2) representation; and that otherwise, ε(ρ) is closed and has empty interior in [inline-graphic 02]. For real characters ρ, we give a complete classification of ε(ρ), and show that ε(ρ) has either 0, 1 or infinitely many elements, and in the last case, ε(ρ) is either a Cantor subset of [inline-graphic 03] or is [inline-graphic 04] itself. We also give a similar> classification for "imaginary" characters where the trace of the commutator is less than 2. Finally, we show that for characters with discrete simple length spectrum (not corresponding to dihedral or SU(2) representations), ε(ρ) is a Cantor subset of [inline-graphic 04] if it contains at least three elements.

Generalized Markoff maps and McShane's identity

We study the (relative) SL ( 2 , C ) character varieties of the one-holed torus and the action of the mapping class group on the (relative) character variety. We show that the subset of characters satisfying two simple conditions called the Bowditch Q-conditions is open in the relative character variety and that the mapping class group acts properly discontinuously on this subset. Furthermore, this is the largest open subset for which this holds. We also show that a generalization of McShane's identity holds for all characters satisfying the Bowditch Q-conditions. Finally, we show that further variations of the McShane–Bowditch identity hold for characters which are fixed by an Anosov element of the mapping class group and which satisfy a relative version of the Bowditch Q-conditions, with applications to identities for incomplete hyperbolic structures on punctured torus bundles over the circle, and also for closed hyperbolic 3-manifolds which are obtained by hyperbolic Dehn surgery on such manifolds.

The affine deformation space of a rank two Schottky group: a picture gallery

The Margulis invariant of an affine hyperbolic element measures the signed Lorentzian displacement along the unique closed geodesic in its class. Given a group of affine transformations of Minkowski spacetime whose linear part is Schottky, the Margulis invariant is a useful tool in determining properness of its action. This paper, based on a presentation given at CGG IV Oostende 2005, describes the affine deformation space of a rank two Schottky group. Several illustrations are included, contrasting the difference between the case of a three-holed sphere and that of a one-holed torus.

The ${\mathrm {SL}(2,{\mathbb C})}$ character variety of a one-holed torus

The ${\mathrm {SL}(2,{\mathbb C})}$ character variety of a one-holed torus

The modular group action on realSL(2)–characters of a one-holed torus

The group Gamma of automorphisms of the polynomial kappa(x,y,z) = x^2 + y^2 + z^2 - xyz -2 is isomorphic to PGL(2,Z) semi-direct product with (Z/2+Z/2). For t in R, Gamma-action on ktR = kappa^{-1}(t) intersect R displays rich and varied dynamics. The action of Gamma preserves a Poisson structure defining a Gamma-invariant area form on each ktR. For t < 2, the action of Gamma is properly discontinuous on the four contractible components of ktR and ergodic on the compact component (which is empty if t < -2). The contractible components correspond to Teichmueller spaces of (possibly singular) hyperbolic structures on a torus M-bar. For t = 2, the level set ktR consists of characters of reducible representations and comprises two ergodic components corresponding to actions of GL(2,Z) on (R/Z)^2 and R^2 respectively. For 2 < t <= 18, the action of Gamma on ktR is ergodic. Corresponding to the Fricke space of a three-holed sphere is a Gamma-invariant open subset Omega subset R^3 whose components are permuted freely by a subgroup of index 6 in Gamma. The level set ktR intersects Omega if and only if t > 18, in which case the Gamma-action on the complement ktR - Omega is ergodic.