Once-punctured Torus Research Articles

Earthquakes on the Once-Punctured Torus

We study earthquake deformations on Teichmüller space associated with simple closed curves of the once-punctured torus. We describe two methods to get an explicit form of the earthquake deformation for any simple closed curve. The first method is rooted in linear recurrence relations, the second in hyperbolic geometry. The two methods align, providing both an algebraic and geometric interpretation of the earthquake deformations. We convert the expressions to other coordinate systems for Teichmüller space to examine earthquake deformations further. Two families of curves are used as examples. Examining the limiting behavior of each gives insight into earthquakes about measured geodesic laminations, of which simple closed curves are a special case.

Music of moduli spaces

A musical instrument, the plastic hormonica, is defined here as a birthday present for Dennis Sullivan, who pioneered and helped popularize the hyperbolic geometry underlying its construction. This plastic hormonica is based upon the Farey tessellation of the Poincaré disk decorated by its standard osculating horocycles centered at the rationals. In effect, one taps or holds points of another tessellation \tau with the same decorating horocycles to produce sounds depending on the fact that the lambda length of e\in\tau with this decoration is always an integer. Explicitly, tapping a decorated edge e\in\tau with lambda length \lambda produces a tone of frequency 440\, \xi^{\lambda-12N} , where \xi^{12}=2 and N is some fixed positive integer shift of octave. Another type of tap on edges of \tau is employed to apply flips, which may be equivariant for a Fuchsian group preserving \tau . Sounding the frequency for the edge after an equivariant flip, one can thereby audibly experience paths in Riemann moduli spaces and listen to mapping classes. The resulting chords, which arise from an ideal triangle complementary to \tau by sounding the frequencies of its frontier edges, correspond to a generalization of the classical Markoff triples, which are precisely the chords that arise from the once-punctured torus. In the other direction, one can query the genera of specified musical pieces.

On cusps of hyperbolic once-punctured torus bundles over the circle

The geometry of certain canonical triangulation of once-punctured torus bundles over the circle is applied to the problem of computing their cusp tori. We are also concerned with the problem of finding the limit points of the set formed by such cusp tori, inside the moduli space of the torus. Our discussion generalizes examples which were elaborated by H. Helling (unpublished) and F. Guéritaud.

Secondary representation stability and the once-punctured torus

We study stability patterns in the homology groups of the ordered configuration space of the once-punctured torus. Church and Church–Ellenberg–Farb proved the homology groups of the ordered configuration space of a connected noncompact orientable manifold stabilize in a representation theoretic sense as the number of points in the configuration grows. Miller and Wilson proved that there is a secondary representation stability pattern among the unstable homology classes when viewed as FIM+-modules. We prove that the sequence of minimal generators in the nth-homology of the ordered configuration space of 2n−2 points on the once-punctured torus is neither “free” nor “stably-zero” as an FIM+-module, and that this sequence is generated by the homology classes on at most 4 points. This is the first calculation that shows that secondary representation stability is a non-trivial phenomenon in positive-genus surfaces.

Branched Cauchy–Riemann structures on once-punctured torus bundles

Unlike in hyperbolic geometry, the monodromy ideal triangulation of a hyperbolic once-punctured torus bundle $M_f$ has no natural geometric realisation in Cauchy-Riemann (CR) space. By introducing a new type of $3$--cell, we construct a different cell decomposition $\mathcal{D}_f$ of $M_f$ that is always realisable in CR space. As a consequence, we show that every hyperbolic once-punctured torus bundle admits a branched CR structure, whose branch locus is the set of edges of $\mathcal{D}_f$. Furthermore, we explicitly compute the ramification order around each component of the branch locus and analyse the corresponding holonomy representations.

Shapes of hyperbolic triangles and once-punctured torus groups

Let Delta be a hyperbolic triangle with a fixed area varphi . We prove that for all but countably many varphi , generic choices of Delta have the property that the group generated by the pi -rotations about the midpoints of the sides of the triangle admits no nontrivial relations. By contrast, we show for all varphi in (0,pi ){setminus }mathbb {Q}pi , a dense set of triangles does afford nontrivial relations, which in the generic case map to hyperbolic translations. To establish this fact, we study the deformation space mathfrak {C}_theta of singular hyperbolic metrics on a torus with a single cone point of angle theta =2(pi -varphi ), and answer an analogous question for the holonomy map rho _xi of such a hyperbolic structure xi . In an appendix by Gao, concrete examples of theta and xi in mathfrak {C}_theta are given where the image of each rho _xi is finitely presented, non-free and torsion-free; in fact, those images will be isomorphic to the fundamental groups of closed hyperbolic 3-manifolds.

On the dynamics on the SU(2)-character variety of a once-punctured torus

On the dynamics on the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>S</mml:mi><mml:mi>U</mml:mi><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math>-character variety of a once-punctured torus

Divergence, exotic convergence and self-bumping in quasi-Fuchsian spaces

In this paper, we study the topology of the boundaries of quasi-Fuchsian spaces. We first show for a given convergent sequence of quasi-Fuchsian groups, how we can know the end invariant of the limit group from the information on the behaviour of conformal structures at infinity of the groups. This result gives rise to a sufficient condition for divergence of quasi-Fuchsian groups, which generalises Ito's result in the once-punctured torus case to higher genera. We further show that quasi-Fuchsian groups can approach a b-group not along Bers slices only when the limit has isolated parabolic loci. This makes it possible to give a necessary condition for points on the boundaries of quasi-Fuchsian spaces to be self-bumping points. We use model manifolds invented by Minsky and their geometric limits studied by Ohshika-Soma to prove these results. This paper has been made as self-contained as possible so that the reader does not need to consult the paper of Ohshika-Soma directly.

Minimal-Area Metrics on the Swiss Cross and Punctured Torus

The closed string field theory minimal-area problem asks for the conformal metric of least area on a Riemann surface with the condition that all non-contractible closed curves have length at least 2\pi. Through every point in such a metric there is a geodesic that saturates the length condition, and saturating geodesics in a given homotopy class form a band. The extremal metric is unknown when bands of geodesics cross, as it happens for surfaces of non-zero genus. We use recently proposed convex programs to numerically find the minimal-area metric on the square torus with a square boundary, for various sizes of the boundary. For large enough boundary the problem is equivalent to the "Swiss cross" challenge posed by Strebel. We find that the metric is positively curved in the two-band region and flat in the single-band regions. For small boundary the metric develops a third band of geodesics wrapping around it, and has both regions of positive and negative curvature. This surface can be completed to provide the minimal-area metric on a once-punctured torus, representing a closed-string tadpole diagram.

A necessary and sufficient condition for a surface sum of two handlebodies to be a handlebody

The main results of the paper is that we give a characteristics for an annulus sum and a once-punctured torus sum of two handlebodies to be a handlebody as follows: 1. The annulus sum $H=H_1\cup_A H_2$ of two handlebodies $H_1$ and $H_2$ is a handlebody if and only if the core curve of $A$ is a longitude for either $H_1$ or $H_2$. 2. Let $H=H_1\cup_T H_2$ be a surface sum of two handlebodies $H_1$ and $H_2$ along a once-punctured torus $T$. Suppose that $T$ is incompressible in both $H_1$ and $H_2$. Then $H$ is a handlebody if and only if the there exists a collection $\{\delta, \sigma\}$ of simple closed curves on $T$ such that either $\{\delta, \sigma\}$ is primitive in $H_1$ or $H_2$, or $\{\delta\}$ is primitive in $H_1$ and $\{\sigma\}$ is primitive in $H_2$.

COARSE AND FINE GEOMETRY OF THE THURSTON METRIC

We study the geometry of the Thurston metric on the Teichmüller space of hyperbolic structures on a surface$S$. Some of our results on the coarse geometry of this metric apply to arbitrary surfaces$S$of finite type; however, we focus particular attention on the case where the surface is a once-punctured torus. In that case, our results provide a detailed picture of the infinitesimal, local, and global behavior of the geodesics of the Thurston metric, as well as an analogue of Royden’s theorem.

On minimal ideal triangulations of cusped hyperbolic 3‐manifolds

Previous work of the authors studies minimal triangulations of closed 3-manifolds using a characterisation of low degree edges, embedded layered solid torus subcomplexes and 1-dimensional Z 2 -cohomology. The underlying blueprint is now used in the study of minimal ideal triangulations. As an application, it is shown that the monodromy ideal triangulations of the hyperbolic once-punctured torus bundles are minimal.

Cusp excursions for the earthquake flow on the once-punctured torus

In this paper we study the typical speed of a generic earthquake trajectory leaving compact sets in the moduli space of the once-punctured torus. Mirzakhani showed that the earthquake flow is measurably equivalent to the horocyclic flow, which has been studied extensively. Our main result shows that the earthquake flow and the horocyclic flow behave very differently in cusp excursions. In particular, we prove a relation between the systole function and continued fractions and discuss the cusp excursions of earthquake trajectories.

DAHA and skein algebra of surfaces: double-torus knots

We study a topological aspect of rank-1 double affine Hecke algebra (DAHA). Clarified is a relationship between the DAHA of A1-type (resp. CC1-type) and the skein algebra on a once-punctured torus (resp. a 4-punctured sphere), and the SL(2;Z) actions of DAHAs are identified with the Dehn twists on the surfaces. Combining these two types of DAHA, we construct the DAHA representation for the skein algebra on a genus-two surface, and we propose a DAHA polynomial for a double-torus knot, which is a simple closed curve on a genus two Heegaard surface in S3. Discussed is a relationship between the DAHA polynomial and the colored Jones polynomial.

BRANCHED CAUCHY–RIEMANN STRUCTURES ON ONCE-PUNCTURED TORUS BUNDLES

Unlike in hyperbolic geometry, the monodromy ideal triangulation of a hyperbolic once-punctured torus bundle $M_f$ has no natural geometric realisation in Cauchy-Riemann (CR) space. By introducing a new type of $3$--cell, we construct a different cell decomposition $\mathcal{D}_f$ of $M_f$ that is always realisable in CR space. As a consequence, we show that every hyperbolic once-punctured torus bundle admits a branched CR structure, whose branch locus is the set of edges of $\mathcal{D}_f$. Furthermore, we explicitly compute the ramification order around each component of the branch locus and analyse the corresponding holonomy representations.

Profinite Rigidity in the SnapPea Census

A well-known question asks whether any two non-isometric finite volume hyperbolic 3-manifolds are distinguished from each other by the finite quotients of their fundamental groups. At present, this has been proved only when one of the manifolds is a once-punctured torus bundle over the circle. We give substantial computational evidence in support of a positive answer, by showing that no two manifolds in the SnapPea census of 72,942 finite volume hyperbolic 3-manifolds have the same finite quotients.

Note on Character Varieties and Cluster Algebras

We use Bonahon-Wong's trace map to study character varieties of the once-punctured torus and of the 4-punctured sphere. We clarify a relationship with cluster algebra associated with ideal triangulations of surfaces, and we show that the Goldman Poisson algebra of loops on surfaces is recovered from the Poisson structure of cluster algebra. It is also shown that cluster mutations give the automorphism of the character varieties. Motivated by a work of Chekhov-Mazzocco-Rubtsov, we revisit confluences of punctures on sphere from cluster algebraic viewpoint, and we obtain associated affine cubic surfaces constructed by van der Put-Saito based on the Riemann-Hilbert correspondence. Further studied are quantizations of character varieties by use of quantum cluster algebra.

Tessellating the Moduli Space of Strictly Convex Projective Structures on the Once-Punctured Torus

ABSTRACTWe show that associating the Euclidean cell decomposition due to Cooper and Long to each point of the moduli space of marked strictly convex real projective structures of finite volume on the once-punctured torus gives this moduli space a natural cell decomposition. The proof makes use of coordinates due to Fock and Goncharov, the action of the mapping class group as well as algorithmic real algebraic geometry. We also show that the decorated moduli space of marked strictly convex real projective structures of finite volume on the thrice-punctured sphere has a natural cell decomposition.

Presentations of Roger and Yang’s Kauffman bracket arc algebra

We provide a presentation of the Roger and Yang's Kauffman bracket arc algebra for the once-punctured torus and punctured spheres with three or fewer punctures.

Commensurability between once-punctured torus groups and once-punctured Klein bottle groups

Commensurability between once-punctured torus groups and once-punctured Klein bottle groups