Action Of Mapping Class Group Research Articles

Spherical objects, transitivity and auto-equivalences of Kodaira cycles via gentle algebras

This paper studies the class of spherical objects over any Kodaira n -cycle of projective lines and provides a parametrization of their isomorphism classes in terms of closed curves on the n -punctured torus without self-intersections. Employing recent results on gentle algebras, we derive a topological model for the bounded derived category of any Kodaira cycle. The groups of triangle auto-equivalences of these categories are computed and are shown to act transitively on isomorphism classes of spherical objects. This answers a question by Polishchuk (2002) and extends earlier results by Burban–Kreußler (2012) and Lekili–Polishchuk (2019). The description of auto-equivalences is further used to establish faithfulness of a mapping class group action defined by Sibilla (2014). The final part describes the closed curves which correspond to vector bundles and simple vector bundles. This leads to an alternative proof of a result by Bodnarchuk–Drozd–Greuel (2012) which states that simple vector bundles on cycles of projective lines are uniquely determined by their multi-degree, rank and determinant. As a by-product, we obtain a closed formula for the cyclic sequence of any simple vector bundle on C_{n} as introduced by Burban–Drozd–Greuel (2001).

Random walks on mapping class groups

This survey is concerned with random walks on mapping class groups. We illustrate how the actions of mapping class groups on Teichmüller spaces or curve complexes reveal the nature of random walks and vice versa. Our emphasis is on the analogues of classical theorems such as laws of large numbers and central limit theorems and the properties of harmonic measures under optimal moment conditions. We also explain the geometric analogy between Gromov hyperbolic spaces and Teichmüller spaces that has been used to copy the properties of random walks from one to the other.

Two simultaneous actions of big mapping class groups

We study two actions of big mapping class groups. The first is an action by isometries on a Gromov-hyperbolic graph. The second is an action by homeomorphisms on a circle in which the vertices of the graph naturally embed. The first two parts of the paper are devoted to the definition of objects and tools needed to introduce these two actions; in particular, we define and prove the existence of equators for infinite type surfaces, we define the hyperbolic graph and the circle needed for the actions, and we describe the Gromov-boundary of the graph using the embedding of its vertices in the circle. The third part focuses on some fruitful relations between the dynamics of the two actions. For example, we prove that loxodromic elements (for the first action) necessarily have rational rotation number (for the second action). In addition, we are able to construct non-trivial quasimorphisms on many subgroups of big mapping class groups, even if they are not acylindrically hyperbolic.

Applications of the algebraic geometry of the Putman–Wieland conjecture

AbstractWe give two applications of our prior work toward the Putman–Wieland conjecture. First, we deduce a strengthening of a result of Marković–Tošić on virtual mapping class group actions on the homology of covers. Second, let and let be a finite ‐cover of topological surfaces. We show the virtual action of the mapping class group of on an ‐isotypic component of has nonunitary image.

Ergodicity of the mapping class group action on Deroin–Tholozan representations

This note investigates the dynamics of the mapping class group action on compact connected components of relative character varieties of surface group representations into PSL(2, \mathbb{R} ), discovered by Deroin and Tholozan. We apply symplectic methods developed by Goldman and Xia to prove that the action is ergodic.

Weakly framed surface configurations, Heisenberg homology, and mapping class group action

We obtain representations of the \(\texttt{f}\)-based mapping class group of oriented punctured surfaces from an action of mapping classes on Heisenberg homologies of a circle bundle over surface configurations.

Global fixed points of mapping class group actions and a theorem of Markovic

We give a short and elementary proof of the non-realizability of the mapping class group via homeomorphisms. This was originally established by Markovic, resolving a conjecture of Thurston. With the tools established in this paper, we also obtain some rigidity results for actions of the mapping class group on Euclidean spaces.

Graphs of curves for surfaces with finite-invariance index \(1\)

In this note we make progress toward a conjecture of Durham–Fanoni–Vlamis, showing that every infinite-type surface with fi­ni­te-invariance index \(1\) and no nondisplaceable compact subsurfaces fails to have a good graph of curves, that is, a connected graph where vertices represent homotopy classes of essential simple closed curves and with a natural mapping class group action having infinite diameter orbits. Our arguments use tools developed by Mann–Rafi in their study of the coarse geometry of big mapping class groups.

Virtual Critical Regularity of Mapping Class Group Actions on the Circle

We show that if G1 and G2 are non-solvable groups, then no C1,τ action of \((G_{1}\times G_{2})*\mathbb {Z}\) on S1 is faithful for τ > 0. As a corollary, if S is an orientable surface of complexity at least three then the critical regularity of an arbitrary finite index subgroup of the mapping class group Mod(S) with respect to the circle is at most one, thus strengthening a result of the first two authors with Baik.

The symplectic fermion ribbon quasi-Hopf algebra and the [formula omitted]-action on its centre

We introduce a family of factorisable ribbon quasi-Hopf algebras Q(N) for N a positive integer: as an algebra, Q(N) is the semidirect product of CZ2 with the direct sum of a Graßmann and a Clifford algebra in 2N generators. We show that RepQ(N) is ribbon equivalent to the symplectic fermion category SF(N) that was computed in [54] from conformal blocks of the corresponding logarithmic conformal field theory. The latter category in turn is conjecturally ribbon equivalent to representations of Vev, the even part of the symplectic fermion vertex operator super algebra.Using the formalism developed in [20] we compute the projective SL(2,Z)-action on the centre of Q(N) as obtained from Lyubashenko's general theory of mapping class group actions for factorisable finite ribbon categories. This allows us to test a conjectural non-semisimple version of the modular Verlinde formula: we verify that the SL(2,Z)-action computed from Q(N) agrees projectively with that on pseudo trace functions of Vev.

Mapping class group actions from Hopf monoids and ribbon graphs

We show that any pivotal Hopf monoid H in a symmetric monoidal category \mathcal{C} gives rise to actions of mapping class groups of oriented surfaces of genus g\geq 1 with n\geq 1 boundary components. These mapping class group actions are given by group homomorphisms into the group of automorphisms of certain Yetter–Drinfeld modules over H . They are associated with edge slides in embedded ribbon graphs that generalise chord slides in chord diagrams. We give a concrete description of these mapping class group actions in terms of generating Dehn twists and defining relations. For the case where \mathcal{C} is finitely complete and cocomplete, we also obtain actions of mapping class groups of closed surfaces by imposing invariance and coinvariance under the Yetter–Drinfeld module structure.

Homotopy coherent mapping class group actions and excision for Hochschild complexes of modular categories

Given any modular category C over an algebraically closed field k, we extract a sequence (Mg)g≥0 of C-bimodules and show that the Hochschild chain complex CH(C;Mg) of C with coefficients in Mg carries a canonical homotopy coherent projective action of the mapping class group of the surface of genus g+1. The ordinary Hochschild complex of C corresponds to CH(C;M0).This result is obtained as part of the following more comprehensive topological structure: We construct a symmetric monoidal functor FC:C-Surfc⟶Chk with values in chain complexes over k defined on a symmetric monoidal category of surfaces whose boundary components are labeled with projective objects in C. The functor FC satisfies an excision property which is formulated in terms of homotopy coends. In this sense, any modular category gives naturally rise to a modular functor with values in chain complexes. In zeroth homology, it recovers Lyubashenko's mapping class group representations.The chain complexes in our construction are explicitly computable by choosing a marking on the surface, i.e. a cut system and a certain embedded graph. For our proof, we replace the connected and simply connected groupoid of cut systems that appears in the Lego-Teichmüller game by a contractible Kan complex.

CFT Correlators for Cardy Bulk Fields via String-Net Models

We show that string-net models provide a novel geometric method to construct invariants of mapping class group actions. Concretely, we consider string-net models for a modular tensor category C. We show that the datum of a specific commutative symmetric Frobenius algebra in the Drinfeld center Z(C) gives rise to invariant string-nets. The Frobenius algebra has the interpretation of the algebra of bulk fields of the conformal field theory in the Cardy case.

Establishing strongly-coupled 3D AdS quantum gravity with Ising dual using all-genus partition functions

We study 3D pure Einstein quantum gravity with negative cosmological constant, in the regime where the AdS radius l is of the order of the Planck scale. Specifically, when the Brown-Henneaux central charge c = 3l/2GN (GN is the 3D Newton constant) equals c = 1/2, we establish duality between 3D gravity and 2D Ising conformal field theory by matching gravity and conformal field theory partition functions for AdS spacetimes with general asymptotic boundaries. This duality was suggested by a genus-one calculation of Castro et al. [Phys. Rev. D85 (2012) 024032]. Extension beyond genus-one requires new mathematical results based on 3D Topological Quantum Field Theory; these turn out to uniquely select the c = 1/2 theory among all those with c < 1, extending the previous results of Castro et al. Previous work suggests the reduction of the calculation of the gravity partition function to a problem of summation over the orbits of the mapping class group action on a “vacuum seed”. But whether or not the summation is well-defined for the general case was unknown before this work. Amongst all theories with Brown-Henneaux central charge c < 1, the sum is finite and unique only when c = 1/2, corresponding to a dual Ising conformal field theory on the asymptotic boundary.

Rigidity of mapping class group actions on S1

The mapping class group $\mathrm{Mod}_{g, 1}$ of a surface with one marked point can be identified with an index two subgroup of $\mathrm{Aut}(\pi_1 \Sigma_g)$. For a surface of genus $g \geq 2$, we show that any action of $\mathrm{Mod}_{g, 1}$ on the circle is either semi-conjugate to its natural action on the Gromov boundary of $\pi_1 \Sigma_g$, or factors through a finite cyclic group. For $g \geq 3$, all finite actions are trivial. This answers a question of Farb.

Extended TQFTs via generators and relations I: The extended toric code

In his PhD thesis [G. Goosen, Oriented 123-tqfts via string-nets and state-sums, PhD thesis, Stellenbosch University, Stellenbosch (2018)], Goosen combined the string-net and the generators-and-relations formalisms for arbitrary once-extended 3-dimensional topological quantum field theories (TQFTs). In this paper, we work this out in detail for the simplest nontrivial example, where the underlying spherical fusion category is the category of [Formula: see text]-graded vector spaces. This allows us to give an elementary string-net description of the linear maps associated to 3-dimensional bordisms. The string-net formalism also simplifies the description of the mapping class group action in the resulting TQFT. We conclude the paper by performing some example calculations from this viewpoint.

The mapping class group action on -character varieties

AbstractLet $\unicode[STIX]{x1D6F4}$ be a compact orientable surface of genus $g=1$ with $n=1$ boundary component. The mapping class group $\unicode[STIX]{x1D6E4}$ of $\unicode[STIX]{x1D6F4}$ acts on the $\mathsf{SU}(3)$-character variety of $\unicode[STIX]{x1D6F4}$. We show that the action is ergodic with respect to the natural symplectic measure on the character variety.

Monadic cointegrals and applications to quasi-Hopf algebras

For $\mathcal{C}$ a finite tensor category we consider four versions of the central monad, $A_1, \dots, A_4$ on $\mathcal{C}$. Two of them are Hopf monads, and for $\mathcal{C}$ pivotal, so are the remaining two. In that case all $A_i$ are isomorphic as Hopf monads. We define a monadic cointegral for $A_i$ to be an $A_i$-module morphism $\mathbf{1} \to A_i(D)$, where $D$ is the distinguished invertible object of $\mathcal{C}$. We relate monadic cointegrals to the categorical cointegral introduced by Shimizu (2019), and, in case $\mathcal{C}$ is braided, to an integral for the braided Hopf algebra $\mathcal{L} = \int^X X^\vee \otimes X$ in $\mathcal{C}$ studied by Lyubashenko (1995). Our main motivation stems from the application to finite dimensional quasi-Hopf algebras $H$. For the category of finite-dimensional $H$-modules, we relate the four monadic cointegrals (two of which require $H$ to be pivotal) to four existing notions of cointegrals for quasi-Hopf algebras: the usual left/right cointegrals of Hausser and Nill (1994), as well as so-called $\gamma$-symmetrised cointegrals in the pivotal case, for $\gamma$ the modulus of $H$. For (not necessarily semisimple) modular tensor categories $\mathcal{C}$, Lyubashenko gave actions of surface mapping class groups on certain Hom-spaces of $\mathcal{C}$, in particular of $SL(2,\mathbb{Z})$ on $\mathcal{C}(\mathcal{L},\mathbf{1})$. In the case of a factorisable ribbon quasi-Hopf algebra, we give a simple expression for the action of $S$ and $T$ which uses the monadic cointegral.

The Alexander Method for Infinite-Type Surfaces

We prove that for any infinite-type orientable surface S, there exists a collection of essential curves Γ in S such that any homeomorphism that preserves the isotopy classes of the elements of Γ is isotopic to the identity. The collection Γ is countable and has an infinite complement in C(S), the curve complex of S. As a consequence, we obtain that the natural action of the extended mapping class group of S on C(S) is faithful.

Derived equivalences of gentle algebras via Fukaya categories

Following the approach of Haiden–Katzarkov–Kontsevich (Publ Math Inst Hautes Études Sci 126:247–318, 2017), to any homologically smooth mathbb {Z}-graded gentle algebra A we associate a triple (Sigma _A, Lambda _A; eta _A), where Sigma _A is an oriented smooth surface with non-empty boundary, Lambda _A is a set of stops on partial Sigma _A and eta _A is a line field on Sigma _A, such that the derived category of perfect dg-modules of A is equivalent to the partially wrapped Fukaya category of (Sigma _A, Lambda _A ;eta _A). Modifying arguments of Johnson and Kawazumi, we classify the orbit decomposition of the action of the (symplectic) mapping class group of Sigma _A on the homotopy classes of line fields. As a result we obtain a sufficient criterion for homologically smooth graded gentle algebras to be derived equivalent. Our criterion uses numerical invariants generalizing those given by Avella–Alaminos–Geiss in Avella et al. (J Pure Appl Algebra 212(1):228–243, 2008), as well as some other numerical invariants. As an application, we find many new cases when the AAG-invariants determine the derived Morita class. As another application, we establish some derived equivalences between the stacky nodal curves considered in Lekili and Polishchuk (J Topology 11:615–444, 2018)