Let $Σ_{g,b}$ denote a closed orientable surface of genus $g$ with $b$ punctures and let $\rm Mod(Σ_{\textit{g,b}})$ denote its mapping class group. In [Luo] Luo proved that if the genus is at least 3, $\rm Mod(Σ_{\textit{g,b}})$ is generated by involutions. He also asked if there exists a universal upper bound, independent of genus and the number of punctures, for the number of torsion elements/involutions needed to generate $\rm Mod(Σ_{\textit{g,b}})$. Brendle and Farb [BF] gave an answer in the case of $g\geq 3, b=0$ and $g\geq 4, b=1$, by describing a generating set consisting of 6 involutions. Kassabov showed that for every $b$ $\rm Mod(Σ_{\textit{g,b}})$ can be generated by 4 involutions if $g\geq 8$, 5 involutions if $g\geq 6$ and 6 involutions if $g\geq 4$. We proved that for every $b$ $\rm Mod(Σ_{\textit{g,b}})$ can be generated by 4 involutions if $g\geq 7$ and 5 involutions if $g\geq 5$.
Let Mod(Sigma_{g, p}) denote the mapping class group of a connected orientable surface of genus g with p punctures. For every even integer p \geq 10 and g \geq 14, we prove that Mod(Sigma_{g, p}) can be generated by three involutions. If the number of punctures p is odd and \geq 9, we show that Mod(Sigma_{g, p}) for g \geq 13 can be generated by four involutions. Moreover, we show that for an even integer p \geq 4 and 3 \leq g \geq 6, Mod(Sigma_{g, p}) can be generated by four involutions.
Let $N_{g,n}$ denote the closed non-orientable surface of genus $g$ with $n$ punctures and let ${\mathcal N}_{g,n}$ denote the mapping class group of $N_{g,n}$. Szepietowski showed that ${\mathcal N}_{g,n}$ is generated by finitely many involutions. The number of elements in his generating set depends linearly on $g$ and $n$. In the case of $n=0$, Szepietowski found an involution generating set in such a way that the number of its elements does not depend on $g$, showing that ${\mathcal N}_{g,0}$ is generated by four involutions. In this thesis, for $n \geq 0$, we prove that ${\mathcal N}_{g,n…
Two related constructions are studied: (1) The diagonal complex $\mathcal{D}$ and its barycentric subdivision $\mathcal{BD}$ related to a \textit{punctured} oriented surface $F$ equipped with a number of labeled marked points. (2) The symmetric diagonal complex $\mathcal{D}^{inv}$ and its barycentric subdivision $\mathcal{BD}^{inv}$ related to a symmetric (=with an involution) oriented surface $F$ equipped with a number of (symmetrically placed) labeled marked points. Eliminating a puncture gives rise to a bundle whose fibers are homeomorphic to a surgery of the surface $F$. The bundle can be …
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.
Let $\textrm{Mod}(N_{g, p})$ denote the mapping class group of a nonorientable surface of genus $g$ with $p$ punctures. For $g\geq14$, we show that $\textrm{Mod}(N_{g, p})$ can be generated by five elements or by six involutions.
Let $Σ_{g,p}$ be a closed oriented surface of genus $g\geq 1$ with $p$ punctures. Let $\rm Mod(Σ_{\textit{g,p}})$ be the mapping class group of $Σ_{g,p}$. Wajnryb proved in [Wa] that for $p=0, 1$ $\rm Mod({Σ_{\textit{g,p}}})$ is generated by two elements. Korkmaz proved in [Ko] that one of these generators can be taken as a Dehn twist. For $p\geq 2$, We proved that $\rm Mod(Σ_{\textit{g,p}})$ is generated by three elements.
We calculate the virtually-cyclic dimension of the mapping class group of a sphere with at most six punctures. As an immediate consequence, we obtain the virtually-cyclic dimension of the mapping class group of the twice-holed torus and of the closed genus-two surface.
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.
In this paper we show that the normal closure of the mth power of a half-twist has infinite index in the mapping class group of a punctured sphere. Furthermore, in some cases we prove that the quotient of the mapping class group of the punctured sphere by the normal closure of a power of a half-twist contains a free abelian subgroup. As a corollary we prove that the quotient of the hyperelliptic mapping class group of a surface of genus at least two by the normal closure of the mth power of a Dehn twist has infinite order, and for some integers m the quotient contains a free nonabelian subgrou…
Let $Σ$ be a complete finite-area orientable hyperbolic surface with one cusp, and let $\mathcal{R}$ be the space of complete geodesic rays in $Σ$ emanating from the puncture. Then there is a natural action of the mapping class group of $Σ$ on $\mathcal{R}$. We show that this action is "almost everywhere" wandering.
Consider an oriented compact surface F of positive genus, possibly with boundary, and a finite set P of punctures in the interior of F, and define the punctured mapping class group of F relatively to P to be the group of isotopy classes of orientation-preserving homeomorphisms h: F-->F which pointwise fix the boundary of F and such that h(P) = P. In this paper, we calculate presentations for all punctured mapping class groups. More precisely, we show that these groups are isomorphic with quotients of Artin groups by some relations involving fundamental elements of parabolic subgroups.
Let $Σ_{g,p}$ be a oriented connected surface of genus $g$ with $p$ punctures. We denote by $\mathcal{M}_{g,p}$ and $\mathcal{M}_{g,p}^\pm$ the mapping class group and the extended mapping class group of $Σ_{g,p}$, respectively. In this paper, we show that $\mathcal{M}_{g,p}$ and $\mathcal{M}_{g,p}^\pm$ are generated by two element for $g\geq 3$ and $p\geq 0$.
Let Mod_{g,b} denote the mapping class group of a surface of genus g with b punctures. Feng Luo asked in a recent preprint if there is a universal upper bound, independent of genus, for the number of torsion elements needed to generate Mod_{g,b}. We answer Luo's question by proving that 3 torsion elements suffice to generate Mod_{g,0}. We also prove the more delicate result that there is an upper bound, independent of genus, not only for the number of torsion elements needed to generate Mod_{g,b} but also for the order of those elements. In particular, our main result is that 6 involutions (i.…
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 prove that many normal subgroups of the extended mapping class group of a surface with punctures are geometric, that is, that their automorphism groups and abstract commensurator groups are isomorphic to the extended mapping class group. In order to apply our theorem to a normal subgroup we require that the "minimal supports" of its elements satisfy a certain complexity condition that is easy to check in practice. The key ingredient is proving that the automorphism groups of many simplicial complexes associated to punctured surfaces are isomorphic to the extended mapping class group. This r…
We give a presentation for the baseleaf preserving mapping class group $Mod(§)$ of the punctured solenoid $§$. The generators for our presentation were introduced previously, and several relations among them were derived. In addition, we show that $Mod(§)$ has no non-trivial central elements. Our main tool is a new complex of triangulations of the disk upon which $Mod(§)$ acts.
Let $Σ_{g,b}$ denote a closed oriented surface genus $g$ with $b$ punctures and let $Mod_{g,b}$ denote its mapping class group. Luo proved that if the genus is at least 3, the group $Mod_{g,b}$ is generated by involutions. He also asked if there exists a universal upper bound, independent of genus and the number of punctures, for the number of torsion elements/involutions needed to generate $Mod_{g,b}$. Brendle and Farb gave a partial answer in the case of closed surfaces and surfaces with one puncture, by describing a generating set consisting of 7 involutions. Our main result generalizes the…
We develop an algebraic representation for (1,1)-knots using the mapping class group of the twice punctured torus MCG(T,2). We prove that every (1,1)-knot in a lens space L(p,q) can be represented by the composition of an element of a certain rank two free subgroup of MCG(T,2) with a standard element only depending on the ambient space. As a notable examples, we obtain a representation of this type for all torus knots and for all two-bridge knots. Moreover, we give explicit cyclic presentations for the fundamental groups of the cyclic branched coverings of torus knots of type (k,ck+2).
We prove that, for $g\geq19$ the mapping class group of a nonorientable surface of genus $g$, $\textrm{Mod}(N_g)$, can be generated by two elements, one of which is of order $g$. We also prove that for $g\geq26$, $\textrm{Mod}(N_g)$ can be generated by three involutions if $g\geq26$.
We investigate the relation between the topological entropy of pseudo-Anosov maps on surfaces with punctures and the rank of the first homology of their mapping tori. On the surface $S$ of genus $g$ with $n$ punctures, we show that the entropy of a pseudo-Anosov map is bounded from above by $\dfrac{(k+1)\log(k+3)}{|χ(S)|}$ up to a constant multiple when the rank of the first homology of the mapping torus is $k+1$ and $k, g, n$ satisfy a certain assumption. This is a partial generalization of precedent works of Tsai and Agol-Leininger-Margalit.
Let $N_{g}$ denote the closed non-orientable surface of genus $g$ and let ${\mathcal M} _g$ denote the mapping class group of $N_{g}$. Let ${\mathcal T} _g$ denote the twist subgroup of ${\mathcal M} _g$ which is the subgroup of ${\mathcal M} _g$ is generated by all Dehn twists. In this thesis, we proved that ${\mathcal T} _g$ is generated by six involutions for $g \geq 16$ or $g = 14$.
We prove that the extended mapping class group, $\rm Mod^{*}(Σ_{g})$, of a connected orientable surface of genus $g$, can be generated by three involutions for $g\geq 5$. In the presence of punctures, we prove that $\rm Mod^{*}(Σ_{g,p})$ can be generated by three involutions for $g\geq 10$ and $p\geq 6$ (with the exception that for $g\geq 11$, $p$ should be at least $15$).
The positive Dehn twist expressions for the generalizations of the new involutions described in math.GT/0404310 are presented. The homeomorphism types of the Lefschetz fibrations that they define are determined for several examples.
Let X_n be a cycle of n projective lines, and T_n a symplectic torus with n punctures. In this paper we review results appeared in arXiv:1103.2462 and in arXiv:1109.6615, which establish a version of homological mirror symmetry relating X_n and T_n, and define on D^b(Coh(X_n)) an action of the pure mapping class group of T_n.
We study the maximal entropy per unit generator of push-point mapping classes on the punctured disk. Our work is motivated by fluid mixing by rods in a planar domain. If a single rod moves among N-fixed obstacles, the resulting fluid diffeomorphism is in the push-point mapping class associated with the loop in π_1(D^2 - {N points}) traversed by the single stirrer. The collection of motions in each of which the stirrer goes around a single obstacle generate the group of push-point mapping classes, and the entropy efficiency with respect to these generators gives a topological measure of the mix…
We construct a certain cross product of two copies of the braided dual $\tilde H$ of a quasitriangular Hopf algebra $H$, which we call the elliptic double $E_H$, and which we use to construct representations of the punctured elliptic braid group extending the well-known representations of the planar braid group attached to $H$. We show that the elliptic double is the universal source of such representations. We recover the representations of the punctured torus braid group obtained in arXiv:0805.2766, and hence construct a homomorphism to the Heisenberg double $D_H$, which is an isomorphism if…
We introduce subgroups ${\mathcal{B}}_g< {\mathcal H}_g$ of the mapping class group $Mod(Σ_g)$ of a closed surface of genus $g \ge 0$ with a Cantor set removed, which are extensions of Thompson's group $V$ by a direct limit of mapping class groups of compact surfaces of genus $g$. We first show that both ${\mathcal{B}}_g$ and ${\mathcal H}_g$ are finitely presented, and that ${\mathcal H}_g$ is dense in $Mod(Σ_g)$. We then exploit the relation with Thompson's groups to study properties ${\mathcal B}_g$ and ${\mathcal H}_g$ in analogy with known facts about finite-type mapping class groups. For…
The well-known fact that any genus $g$ symplectic Lefschetz fibration $ X^{4}\to S^{2}$ is given by a word that is equal to the identity element in the mapping class group and each of whose elements is given by a positive Dehn twist, provides an intimate relationship between words in the mapping class group and 4-manifolds that are realized as symplectic Lefschetz fibrations. In this article we provide new words in the mapping class group, hence new symplectic Lefschetz fibrations. We also compute the signatures of those symplectic Lefschetz fibrations.
We investigate the problem of when big mapping class groups are generated by involutions. Restricting our attention to the class of self-similar surfaces, which are surfaces with self-similar ends space, as defined by Mann and Rafi, and with 0 or infinite genus, we show that, when the set of maximal ends is infinite, then the mapping class groups of these surfaces are generated by involutions, normally generated by a single involution, and uniformly perfect. In fact, we derive this statement as a corollary of the corresponding statement for the homeomorphism groups of these surfaces. On the ot…
In this paper we provide a classification of fundamental group elements representing simple closed curves on the punctured Klein bottle, Similar to the Birman-Series classification of curves on the punctured torus[1]. In the process, an explicit description of the mapping class group is given. We then apply this to give a counterexample the simple loop conjecture for representations from the Klein bottle group to PGL(2,R).
The mapping class group $Γ^k(N_g)$ of a non-orientable surface with punctures is studied via classical homotopy theory of configuration spaces. In particular, we obtain a non-orientable version of the Birman exact sequence. In the case of $\mathbb R {\rm P}^2$, we analize the Serre spectral sequence of a fiber bundle $F_k(\mathbb R {\rm P}^2)/Σ_k \to X_k \to BSO(3)$ where $X_k$ is a $K(Γ^k(\mathbb R {\rm P}^2),1)$ and $F_k(\mathbb R {\rm P}^2)/Σ_k$ denotes the configuration space of unordered $k$-tuples of distinct points in $\mathbb R {\rm P}^2$. As a consequence, we express the mod-2 cohomol…
We use quantum invariants to define an analytic family of representations for the mapping class group of a punctured surface. The representations depend on a complex number A with |A| <= 1 and act on an infinite-dimensional Hilbert space. They are unitary when A is real or imaginary, bounded when |A|<1, and only densely defined when |A| = 1 and A is not a root of unity. When A is a root of unity distinct from 1, -1, i, -i the representations are finite-dimensional and isomorphic to the "Hom" version of the well-known TQFT quantum representations. The unitary representations in the interval [-1…
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.
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.
We prove that both the hyperelliptic mapping class group and the extended hyperelliptic mapping class group are generated by two torsion elements. We also compute the index of the subgroup of the hyperelliptic mapping class group which is generated by involutions and we prove that the extended hyperelliptic mapping class group is generated by three orientation reversing involutions.
A fast algorithm for counting intersections of two normal curves on a triangulated surface is proposed. It yields a convenient way for treating mapping class groups of punctured surfaces by presenting mapping classes by matrices, and the composition by an exotic matrix multiplication. An efficient solution of the word problem for mapping class groups of punctured surfaces is proposed, with efficiency understood in a more restrictive way than the most common one.
The braid groups B_n can be defined as the mapping class group of the n-punctured disc. The Lawrence-Krammer representation of the braid group B_n is the induced action on a certain twisted second homology of the space of unordered pairs of points in the n-punctured disc. Recently, Daan Krammer showed that this is a faithful representation in the case n=4. In this paper, we show that it is faithful for all n.
For a nonorientable surface, the twist subgroup is an index 2 subgroup of the mapping class group. It is generated by Dehn twists about two-sided simple closed curves. In this paper, we study involution generators of the twist subgroup. We give generating sets of involutions with the smallest number of elements our methods allow.
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 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.
We give counterexamples to a question of Bowditch that if a non-elementary type-preserving representation $ρ:π_1(Σ_{g,n})\rightarrow PSL(2;\mathbb R)$ of a punctured surface group sends every non-peripheral simple closed curve to a hyperbolic element, then must $ρ$ be Fuchsian. The counterexamples come from relative Euler class $\pm1$ representations of the four-punctured sphere group. We also show that the mapping class group action on each non-extremal component of the character space of type-preserving representations of the four-punctured sphere group is ergodic, which confirms a conjectur…
We obtain representations of the f-based Mapping Class Group of oriented punctured surfaces from an action of mapping classes on Heisenberg homologies of a circle bundle over surface configurations.
For a pseudo-Anosov homeomorphism $f$ on a closed surface of genus $g\geq 2$, for which the entropy is on the order $\frac{1}{g}$ (the lowest possible order), Farb-Leininger-Margalit showed that the volume of the mapping torus is bounded, independent of $g$. We show that the analogous result fails for a surface of fixed genus $g$ with $n$ punctures, by constructing pseudo-Anosov homeomorphism with entropy of the minimal order $\frac{\log n}{n}$, and volume tending to infinity.
We study the ideal triangulation graph $T(S)$ of a punctured surface $S$ of finite type. We show that if $S$ is not the sphere with at most three punctures or the torus with one puncture, then the natural map from the extended mapping class group of $S$ into the simplicial automorphism group of $T(S)$ is an isomorphism. We also show that under the same conditions on $S$, the graph $T(S)$ equipped with its natural simplicial metric is not Gromov hyperbolic. Thus, from the point of view of Gromov hyperbolicity, the situation of $T(S)$ is different from that of the curve complex of $S$.
