Narrow Results

The Goeritz group of the standard genus-g Heegaard splitting of the three sphere, $G_g$, acts on the space of isotopy classes of reducing spheres for this Heegaard splitting. Scharlemann [Automorphisms of the 3-sphere that preserve a genus two Heegaard splitting, Bol. Soc. Mat. Mexicana10 (2004) 503–514] uses this action to prove that $G_2$ is finitely generated. In this paper, we give an algorithm to construct any reducing sphere from a standard reducing sphere for a genus-2 Heegaard splitting of the $S^3$. Using this we give an alternate proof of the finite generation of $G_2$ assuming the finite generation of the stabilizer of the standard reducing sphere.

In this paper, we study the normal generation of the mapping class group. We first show that a mapping class is a normal generator if its restriction on the invariant subsurface normally generates the (pure) mapping class group of the subsurface. As an application, we provided a criterion for reducible mapping classes to normally generate the mapping class groups in terms of its asymptotic translation lengths on Teichmüller spaces. This is an analogue to the work of Lanier–Margalit dealing with pseudo-Anosov normal generators.

Let Σ_g be a closed orientable surface of genus g and let Diff₀(Σ_g, area) be the identity component of the group of area-preserving diffeomorphisms of Σ_g. In this paper, we present the extension of Gambaudo–Ghys construction to the case of a closed hyperbolic surface Σ_g, i.e. we show that every nontrivial homogeneous quasi-morphism on the braid group on n strings of Σ_g defines a nontrivial homogeneous quasi-morphism on the group Diff₀(Σ_g, area). As a consequence we give another proof of the fact that the space of homogeneous quasi-morphisms on Diff₀(Σ_g, area) is infinite-dimensional. Let Ham(Σ_g) be the group of Hamiltonian diffeomorphisms of Σ_g. As an application of the above construction we construct two injective homomorphisms Z^m → Ham(Σ_g), which are bi-Lipschitz with respect to the word metric on Z^m and the autonomous and fragmentation metrics on Ham(Σ_g). In addition, we construct a new infinite family of Calabi quasi-morphisms on Ham(Σ_g).

In [5] Grossman showed that outer automorphism groups of free groups and of fundamental groups of compact orientable surfaces are residually finite. In this paper we introduce the concept of "Property E" of groups and show that certain generalized free products and HNN extensions have this property. We deduce that the outer automorphism groups of finitely generated non-triangle Fuchsian groups are residually finite.

Irreducible Artin groups of finite type can be parametrized via their associated Coxeter diagrams into six sporadic examples and four infinite families, each of which is further parametrized by the natural numbers. Within each of these four infinite families, we investigate the relationship between elementary equivalence and isomorphism. For three out of the four families, we show that two groups in the same family are equivalent if and only if they are isomorphic; a positive, but weaker, result is also attained for the fourth family. In particular, we show that two braid groups are elementarily equivalent if and only if they are isomorphic. The ${(\forall \exists )}^1$ fragment suffices to distinguish the elementary theories of the groups in question.

As a consequence of our work, we prove that there are infinitely many elementary equivalence classes of irreducible Artin groups of finite type. We also show that mapping class groups of closed surfaces — a geometric analogue of braid groups — are elementarily equivalent if and only if they are isomorphic.

For a finitely generated group, there are two recent generalizations of the notion of a quasiconvex subgroup of a word-hyperbolic group, namely a stable subgroup and a Morse or strongly quasiconvex subgroup. Durham and Taylor [M. Durham and S. Taylor, Convex cocompactness and stability in mapping class groups, Algebr. Geom. Topol.  15(5) (2015) 2839–2859] defined stability and proved stability is equivalent to convex cocompactness in mapping class groups. Another natural generalization of quasiconvexity is given by the notion of a Morse or strongly quasiconvex subgroup of a finitely generated group, studied recently by Tran [H. Tran, On strongly quasiconvex subgroups, To Appear in Geom. Topol., preprint (2017), arXiv:1707.05581] and Genevois [A. Genevois, Hyperbolicities in CAT (0) cube complexes, preprint (2017), arXiv:1709.08843]. In general, a subgroup is stable if and only if the subgroup is Morse and hyperbolic. In this paper, we prove that two properties of being Morse and stable coincide for a subgroup of infinite index in the mapping class group of an oriented, connected, finite type surface with negative Euler characteristic.

Let H_g be a genus g handlebody and MCG_2n(T_g) be the group of the isotopy classes of orientation preserving homeomorphisms of T_g = ∂H_g, fixing a given set of 2n points. In this paper we study two particular subgroups of MCG_2n(T_g) which generalize Hilden groups defined by Hilden in [Generators for two groups related to the braid groups, Pacific J. Math.59 (1975) 475–486]. As well as Hilden groups are related to plat closures of braids, these generalizations are related to Heegaard splittings of manifolds and to bridge decompositions of links. Connections between these subgroups and motion groups of links in closed 3-manifolds are also provided.

For N ≥ 2, we study a certain sequence 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.

Let g and n be integers at least 2, and let G be the pure braid group with n strands on a closed orientable surface of genus g. We describe any injective homomorphism from a finite index subgroup of G into G. As a consequence, we show that any finite index subgroup of G is co-Hopfian.

Let $S_{0,n}$ be an $n$-punctured sphere. For $n\ge 4$, we construct a sequence ${({{𝒳}}_i)}_{i\in \mathbb{N}}$ of finite rigid sets in the pants graph ${𝒫}(S_{0,n})$ such that ${{𝒳}}_1\subset {{𝒳}}_2\subset \cdots \subset {𝒫}(S_{0,n})$ and ${\bigcup }_{i\ge 1}{{𝒳}}_i={𝒫}(S_{0,n})$.

In this paper, we give two formulae of values of the Casson–Walker invariant of 3-manifolds with genus one open book decompositions; one is a formula written in terms of a framed link of a surgery presentation of such a 3-manifold, and the other is a formula written in terms of a representation of the mapping class group of a 1-holed torus. For the former case, we compute the invariant through the combinatorial calculation of the degree 1 part of the LMO invariant. For the latter case, we construct a representation of a central extension of the mapping class group through the action of the degree 1 part of the LMO invariant on the space of Jacobi diagrams on two intervals, and compute the invariant as the trace of the representation of a monodromy of an open book decomposition.

Let $R$ be a compact, connected, orientable surface of genus $g$ with $n$ boundary components with $g\ge 2$, $n\ge 0$. Let ${𝒩}(R)$ be the nonseparating curve graph, ${𝒞}(R)$ be the curve graph and ${\mathscr{H}}{𝒯}(R)$ be the Hatcher–Thurston graph of $R$. We prove that if $\lambda :{𝒩}(R)\to {𝒩}(R)$ is an edge-preserving map, then $\lambda$ is induced by a homeomorphism of $R$. We prove that if $𝜃:{𝒞}(R)\to {𝒞}(R)$ is an edge-preserving map, then $𝜃$ is induced by a homeomorphism of $R$. We prove that if $R$ is closed and $\tau :{\mathscr{H}}{𝒯}(R)\to {\mathscr{H}}{𝒯}(R)$ is a rectangle preserving map, then $\tau$ is induced by a homeomorphism of $R$. We also prove that these homeomorphisms are unique up to isotopy when $(g,n)\ne (2,0)$.

The Reshetikhin-Turaev representation of the mapping class group of an orientable surface is computed explicitly in the case r = 4. It is then shown that the restriction of this representation to the Torelli group is equal to the sum of the Birman-Craggs homomorphisms. The proof makes use of an explicit correspondence between the basis vectors of the representation space, and the Z/2Z-quadratic forms on the first homology of the surface. This result corresponds to the fact, shown by Kirby and Melvin, that the three-manifold invariant when r = 4 is related to spin structures on the associated four-manifold.

The quantum group construction of Reshetikhin and Turaev provides representations of the mapping class group, indexed by an integer parameter r. This paper presents computations of these representations when r=6, and analyzes their relationship to other topological invariants. It is shown that in genus 2, the representation splits into two summands. The first summand factors through the mapping class group action on the first homology of the surface with Z/3Z coefficients, while the second summand can be analyzed via its restriction to the subgroup of the mapping class group which is normally generated by the sixth power of a Dehn twist on a nonseparating curve. This analysis reveals a connection to the homology intersection pairing on the surface, and also yields information about the kernel and image of the representation. It is also shown that the representation yields a family of 2-dimensional nonabelian representations of the Torelli group.

This paper continues the program established by the author in [Wr] to relate the Reshetikhin-Turaev representations at specific roots of unity to classical invariants.

In [M. De Renzi, A. Gainutdinov, N. Geer, B. Patureau-Mirand and I. Runkel, 3-dimensional TQFTs from non-semisimple modular categories, preprint (2019), arXiv:1912.02063[math.GT]], we constructed 3-dimensional topological quantum field theories (TQFTs) using not necessarily semisimple modular categories. Here, we study projective representations of mapping class groups of surfaces defined by these TQFTs, and we express the action of a set of generators through the algebraic data of the underlying modular category $\mathcal{𝒞}$. This allows us to prove that the projective representations induced from the non-semisimple TQFTs of the above reference are equivalent to those obtained by Lyubashenko via generators and relations in [V. Lyubashenko, Invariants of 3-manifolds and projective representations of mapping class groups via quantum groups at roots of unity, Comm. Math. Phys. 172(3) (1995) 467–516, arXiv:hep-th/9405167]. Finally, we show that, when $\mathcal{𝒞}$ is the category of finite-dimensional representations of the small quantum group of ${𝔰𝔩}_2$, the action of all Dehn twists for surfaces without marked points has infinite order.

In this paper, we consider a certain subgroup ${\mathrm{\text{IA}}}_n^{+}$ of the IA-automorphism group of a free group. We determine the images of the $k$th Johnson homomorphism restricted to ${\mathrm{\text{IA}}}_n^{+}$ for any $k\ge 1$ and $n\ge 2$. By using this result, we give an affirmative answer to the Andreadakis conjecture restricted for ${\mathrm{\text{IA}}}_n^{+}$. Namely, we show that the intersection of the Andreadakis–Johnson filtration and ${\mathrm{\text{IA}}}_n^{+}$ coincides with the lower central series of ${\mathrm{\text{IA}}}_n^{+}$. In a series of this research, we obtain additional results on the integral (co)homology groups of ${\mathrm{\text{IA}}}_n^{+}$. In particular, we determine the first homology group, and study the cup product of first cohomologies of ${\mathrm{\text{IA}}}_n^{+}$. Furthermore, we construct nontrivial second homology classes of ${\mathrm{\text{IA}}}_n^{+}$ by observing its generators and relators, and show that the second cohomology group is not generated by cup products of the first cohomology groups.

In this paper we define a new version of Grothendieck–Teichmüller group defined by three generalized equations coming from finite-order diffeomorphisms, and we prove that it is isomorphic to the known version IΓ of the Grothendieck–Teichmüller defined in [H. Nakamura and L. Schneps, On a subgroup of the Grothendieck–Teichmüller group acting on the tower of profinite Teichmüller modular groups, Invent. Math.141 (2000) 503–560]. We show that acts on the full mapping class groups for 2g - 2 + n > 0. We then prove that the conjugacy classes of prime-order torsion of are exactly the discrete prime-order ones of . Using this we prove that acts on prime-order torsion elements of in a particular way called λ-conjugacy, analogous to the Galois action on inertia.

We show that the mapping class group of a handlebody V of genus at least 2 (with any number of marked points or spots) is exponentially distorted in the mapping class group of its boundary surface ∂V. The same holds true for solid tori V with at least two marked points or spots.

For each pair of integers g ≥ 2 and h ≥ 1, we explicitly construct infinitely many fiber sum and section sum indecomposable genus g surface bundles over genus h surfaces whose total spaces are pairwise homotopy inequivalent.

We prove that the entropy norm on the group of diffeomorphisms of a closed orientable surface of positive genus is unbounded.