This document is a practical guide to computations using an automatic structure for the mapping class group of a once-punctured, oriented surface $S$. We describe a quadratic time algorithm for the word problem in this group, which can be implemented efficiently with pencil and paper. The input of the algorithm is a word, consisting of ``chord diagrams'' of ideal triangulations and elementary moves, which represents an element of the mapping class group. The output is a word called a ``normal form'' that uniquely represents the same group element.
We compute the number of systoles, the shortest simple closed geodesics and 2-systoles, the second shortest simple closed geodesics on hyperbolic surfaces homeomorphic to once-punctured torus and four-punctured sphere.
We determine the non-null homologous knots in lens spaces whose exteriors contain properly embedded once-punctured tori. All such knots arise as surgeries on the Whitehead link and are grid number 1 in their lens spaces. As a corollary, we classify once-punctured torus bundles that admit a lens space filling.
Gang, Kim and Yoon have recently proposed a conjecture on a vanishing identity of adjoint Reidemeister torsions of hyperbolic 3-manifolds with torus boundary, from the viewpoint of wrapped M5-branes. In this paper, we provide infinitely many new supporting examples to this conjecture. These examples come from hyperbolic once-punctured torus bundles. We show that the vanishing identity holds for all hyperbolic once-punctured torus bundles with tunnel number one. We also show the vanishing identity does not hold for any torus knot exteriors.
A generalized torsion in a group, an non-trivial element such that some products of its conjugates is the identity. This is an obstruction for a group being bi-orderable. Though it is known that there is a non bi-orderable group without generalized torsions, it is conjectured that 3-manifold groups without generalized torsions are bi-orderable. In this paper, we find generalized torsions in the fundamental groups of once punctured torus bundles which are not bi-orderable. Our result contains a generalized torsion in a tunnel number two hyperbolic once punctured torus bundle.
We prove the hyperbolization theorem for punctured torus bundles and two-bridge link complements by decomposing them into ideal tetrahedra which are then given hyperbolic structures, following Rivin's volume maximization principle.
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…
In this paper, we determine the canonical polyhedral decomposition of every hyperbolic once-punctured torus bundle over the circle. In fact, we show that the only ideal polyhedral decomposition that is straight in the hyperbolic structure and that is invariant under a certain involution is the ideal triangulation defined by Floyd and Hatcher. Unlike previous work on this problem, the techniques we use are not geometric. Instead, they involve angled polyhedral decompositions, thin position and a version of almost normal surface theory.
We show that associating the Euclidean cell decomposition due to Cooper and Long to each point of the moduli space of framed 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 framed strictly convex real projective structures of finite volume on the thrice-punctured sphere has a natural cell decomposition.
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.
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. E…
We prove Thurston's bending measure conjecture for quasifuchsian once punctured torus groups. The conjecture states that the bending measures of the two components of the convex hull boundary uniquely determine the group.
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 $\{δ, σ\}$ of simple clos…
In this paper we study exceptional Dehn fillings on hyperbolic knot manifolds which contain an essential once-punctured torus. Let $M$ be such a knot manifold and let $β$ be the boundary slope of such an essential once-punctured torus. We prove that if Dehn filling $M$ with slope $α$ produces a Seifert fibred manifold, then $Δ(α,β)\leq 5$. Furthermore we classify the triples $(M; α,β)$ when $\D(α,β)\geq 4$. More precisely, when $\D(α,β)=5$, then $M$ is the (unique) manifold $Wh(-3/2)$ obtained by Dehn filling one boundary component of the Whitehead link exterior with slope -3/2, and $(α, β)$ i…
We describe a class $\mathcal{C}$ of punctured torus bundles such that, for each $M \in \mathcal{C}$, all but finitely many Dehn fillings on $M$ are virtually Haken. We show that $\mathcal{C}$ contains infinitely many commensurability classes, and we give evidence that $\mathcal{C}$ includes representatives of ``most'' commensurability classes of punctured torus bundles. In particular, we define an integer-valued complexity function on monodromies $f$ (essentially the length of the LR-factorization of $f_*$ in $PSL_2(\mathbb{Z})$), and use a computer to show that if the monodromy of $M$ has co…
We obtain new variations of the original McShane identity for those SL(2,C)-representations of the once punctured torus group which satisfy the Bowditch conditions, and also for those fixed up to conjugacy by an Anosov mapping class of the torus and satisfying the relative Bowditch conditions.
In these lecture notes, we combine recent homological methods of Kevin Whyte with older dynamical methods developed by Benson Farb and myself, to obtain a new quasi-isometric rigidity theorem for the mapping class group MCG(S) of a once punctured surface S of genus at least 2: if K is a finitely generated group quasi-isometric to MCG(S) then there is a homomorphism K -> MCG(S) with finite kernel and finite index image. This theorem is joint with Kevin Whyte.
We determine the PSL_2(C) and SL_2(C) character varieties of the once-punctured torus bundles with tunnel number one, i.e. the once-punctured torus bundles that arise from filling one boundary component of the Whitehead link exterior. In particular, we determine `natural' models for these algebraic sets, identify them up to birational equivalence with smooth models, and compute the genera of the canonical components. This enables us to compare dilatations of the monodromies of these bundles with these genera. We also determine the minimal polynomials for the trace fields of these manifolds. Ad…
Recent work of Chinburg, Reid, and Stover has shown that certain arithmetic and algebro-geometric properties of the character variety of a hyperbolic knot complement in the 3-sphere $M=S^3\setminus K$ yields topological and number theoretic information about Dehn fillings of M. Specifically, they show how the study of a certain extension problem for quaternion Azumaya algebras is related to topological invariants associated to these fillings. In this paper, we extend their work to the setting of hyperbolic once punctured torus bundles. Along the way, we exhibit new phenomenon in the relevant e…
In this paper, we use a result of Dahmani to show that the Euler class of some power subgroup (the subgroup normally generated by a fixed power of Dehn twist about a non-separating curve) is trivial inside the mapping class group of once punctured surface.
Let M be a hyperbolic manifold of finite volume which fibers over the circle with fiber a once punctured torus, and let S be an arbitrary incompressible surface in M. We determine the characteristic JSJ-subpair of M-S and show, in particular, that the guts of (M,S) is empty.
We explore several families of flip-graphs, all related to polygons or punctured polygons. In particular, we consider the topological flip-graphs of once-punctured polygons which, in turn, contain all possible geometric flip-graphs of polygons with a marked point as embedded sub-graphs. Our main focus is on the geometric properties of these graphs and how they relate to one another. In particular, we show that the embeddings between them are strongly convex (or, said otherwise, totally geodesic). We also find bounds on the diameters of these graphs, sometimes using the strongly convex embeddin…
To each once-punctured-torus bundle, $T_ϕ$, over the circle with pseudo-Anosov monodromy $ϕ$, there are associated two tessellations of the complex plane: one, $Δ(ϕ)$, is (the projection from $\infty$ of) the triangulation of a horosphere at $\infty$ induced by the canonical decomposition into ideal tetrahedra, and the other, $CW(ϕ)$, is a fractal tessellation given by the Cannon-Thurston map of the fiber group switching back and forth between gray and white each time it passes through $\infty$. In this paper, we study the relation between $Δ(ϕ)$ and $CW(ϕ)$.
We propose a method to compute complex volume of 2-bridge link complements. Our construction sheds light on a relationship between cluster variables with coefficients and canonical decompositions of link complements.
A triangulation of a surface with fixed topological type is called irreducible if no edge can be contracted to a vertex while remaining in the category of simplicial complexes and preserving the topology of the surface. A complete list of combinatorial structures of irreducible triangulations is made by hand for the once-punctured torus, consisting of exactly 297 non-isomorphic triangulations.
We classify right-veering homeomorphisms of the once-punctured torus using the Burau representation of the 3-strand braid group. We show that reducible and periodic mapping classes in B_3 can be identified as right-veering by consideration of the reduced version of the Burau representation. Given any element beta in B_3, we give a method to quickly determine its action on the generators of the fundamental group of the 3-times punctured disk. This action which determines whether beta is right-veering, left-veering, or neither.
In this paper, we construct polynomial growth harmonic maps from once-punctured Riemann surfaces of any finite genus to any even-sided, regular, ideal polygon in the hyperbolic plane. We also establish their uniqueness within a class of maps which differ by exponentially decaying variations. Previously, harmonic maps from once-punctured spheres to the hyperbolic plane have been parameterized by holomorphic quadratic differentials on the complex plane. Our harmonic maps, mapping a genus g>1 domain to a k-sided polygon, correspond to meromorphic quadratic differentials having one pole of order (…
We prove the identity $$ \prod_γ\left(\frac{e^{l(γ)}+1}{e^{l(γ)}-1}\right)^{2h}=\exp\left(\frac{l_1+l_2+l_3}{2}\right), $$ (or $$ \prod_γ\left(\frac{t(γ)^2}{t(γ)^2-4}\right)^h=\frac{t_1+\sqrt{t_1^2-4}}{2}\cdot\frac{t_2+\sqrt{t_2^2-4}}{2}\cdot\frac{t_3+\sqrt{t_3^2-4}}{2} $$ in trace coordinates), where the product is over all simple closed geodesics on the once-punctured torus, $l(γ)=2\operatorname{arccosh}(t(γ)/2)$ is the length of the geodesic, and $l_i$ ($t_i$) are the lengths (traces) of any triple of simple geodesics $\{γ_i\}$ intersecting at a single point. The exponent $h=h(γ;\{γ_i\})$ i…
Let $Δ$ be a hyperbolic triangle with a fixed area $φ$. We prove that for all but countably many $φ$, generic choices of $Δ$ have the property that the group generated by the $π$--rotations about the midpoints of the sides of the triangle admits no nontrivial relations. By contrast, we show for all $φ\in(0,π)\setminus\mathbb{Q}π$, 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 $\mathfrak{C}_θ$ of singular hyperbolic metrics on a torus with a single cone point of angle $θ…
We derive two types of linearity conditions for mapping class groups of orientable surfaces: one for once-punctured surface, and the other for closed surface, respectively. For the once-punctured case, the condition is described in terms of the action of the mapping class group on the deformation space of linear representations of the fundamental group of the corresponding closed surface. For the closed case, the condition is described in terms of the vector space generated by the isotopy classes of essential simple closed curves on the corresponding surface. The latter condition also describe…
We consider the following question: Which parameters in the extension of a rational pleating ray across the boundary of $\Cal M$, the Maskit embedding of the Teichmüller space of once punctured tori correspond to a Kleinian group? Using methods of Keen and Series and Wright we prove a local result, stating that on each rational ray there is a sequence of parameters in $\overline\Bbb H\setminus\Cal M$ accumulating at the boundary point of $\Cal M$ on the ray. These are the unique parameters on the extended $p/q$ ray for which the special word $W_{p/q}$ is a primitive elliptic Möbius transformat…
We show that for a surface S, the subgraph of the pants graph determined by fixing a collection of curves that cut S into pairs of pants, once-punctured tori, and four-times-punctured spheres is totally geodesic. The main theorem resolves a special case of a conjecture made by Aramayona, Parlier, and Shackleton and has the implication that an embedded product of Farey graphs in any pants graph is totally geodesic. In addition, we show that a pants graph contains a convex n-flat if and only if it contains an n-quasi-flat.
Let $M$ be a once-punctured torus bundle over $S^1$ with monodromy $h$. We show that, under certain hypotheses on $h$, "most" Dehn-fillings of $M$ (in some cases all but finitely many) are virtually $\mathbb{Z}$-representable. We apply our results to show that surgeries on the figure-eight knot with even numerator are virtually $\mathbb{Z}$-representable.
Greg McShane introduced a remarkable identity for lengths of simple closed geodesics on the once punctured torus with a complete, finite volume hyperbolic structure. Bowditch later generalized this and gave sufficient conditions for the identity to hold for general type-preserving representations of a free group on two generators Γto SL(2,C). In this note we extend Bowditch's result by giving necessary and sufficient conditions for the identity to hold, and also for the generalized McShane identity to hold for arbitrary (not necessarily type preserving) representations. We also give a version …
For a given $ε>0$, we show that there exist two finite index subgroups of $PSL_2(\mathbb{Z})$ which are $(1+ε)$-quasisymmetrically conjugated and the conjugation homeomorphism is not conformal. This implies that for any $ε>0$ there are two finite regular covers of the Modular once punctured torus $T_0$ (or just the Modular torus) and a $(1+ε)$-quasiconformal between them that is not homotopic to a conformal map. As an application of the above results, we show that the orbit of the basepoint in the Teichmüller space $T(§)$ of the punctured solenoid $§$ under the action of the corresponding Modu…
Let $Σ$ be a hyperbolic surface. We study the set of curves on $Σ$ of a given type, i.e. in the mapping class group orbit of some fixed but otherwise arbitrary $γ_0$. For example, in the particular case that $Σ$ is a once-punctured torus, we prove that the cardinality of the set of curves of type $γ_0$ and of at most length $L$ is asymptotic to $L^2$ times a constant.
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 …
We prove that the word problem in the mapping class group of the once-punctured surface of genus g has complexity O(|w|^2 g for |w| > log(g) where |w| is the length of the word in a (standard) set of generators. The corresponding bound in the case of the closed surface is O(|w|^2 g^2). We also carry out the same methods for the braid groups, and show that this gives a bound which improves the best known bound in this case; namely, the complexity of the word problem in the n-braid group is O(|w|^2 n), for |w| > log n. We state a similar result for mapping class groups of surfaces with several p…
We construct new monomorphisms between mapping class groups of surfaces. The first family of examples injects the mapping class group of a closed surface into that of a different closed surface. The second family of examples are defined on mapping class groups of once-punctured surfaces and have quite curious behaviour. For instance, some pseudo-Anosov elements are mapped to multi-twists. Neither of these two types of phenomena were previously known to be possible although the constructions are elementary.
