Mapping Class Group Of Surface Research Articles

The mapping class group of a nonorientable surface is quasi-isometrically embedded in the mapping class group of the orientation double cover

The mapping class group of a nonorientable surface is quasi-isometrically embedded in the mapping class group of the orientation double cover

The balanced superelliptic mapping class groups are generated by three elements

Abstract The balanced superelliptic mapping class group is the normalizer of the transformation group of the balanced superelliptic covering in the mapping class group of the total surface. We prove that the balanced superelliptic mapping class groups with either one marked point, one boundary component, or no marked points and boundary are generated by three elements. To prove this, we also show that its liftable mapping class groups are also generated by three elements. These generating sets are minimal except for several cases of closed surfaces.

Surface Houghton groups

AbstractFor every $$n\ge 2$$ n ≥ 2 , the surface Houghton group$${\mathcal {B}}_n$$ B n is defined as the asymptotically rigid mapping class group of a surface with exactly n ends, all of them non-planar. The groups $${\mathcal {B}}_n$$ B n are analogous to, and in fact contain, the braided Houghton groups. These groups also arise naturally in topology: every monodromy homeomorphism of a fibered component of a depth-1 foliation of closed 3-manifold is conjugate into some $${\mathcal {B}}_n$$ B n . As countable mapping class groups of infinite type surfaces, the groups $$\mathcal {B}_n$$ B n lie somewhere between classical mapping class groups and big mapping class groups. We initiate the study of surface Houghton groups proving, among other things, that $$\mathcal {B}_n$$ B n is of type $$\text {F}_{n-1}$$ F n - 1 , but not of type $$\text {FP}_{n}$$ FP n , analogous to the braided Houghton groups.

Notes on hyperelliptic mapping class groups

AbstractHyperelliptic mapping class groups are defined either as the centralizers of hyperelliptic involutions inside mapping class groups of oriented surfaces of finite type or as the inverse images of these centralizers by the natural epimorphisms between mapping class groups of surfaces with marked points. We study these groups in a systematic way. An application of this theory is a counterexample to the genus $2$ case of a conjecture by Putman and Wieland on virtual linear representations of mapping class groups. In the last section, we study profinite completions of hyperelliptic mapping class groups: we extend the congruence subgroup property to the general class of hyperelliptic mapping class groups introduced above and then determine the centralizers of multitwists and of open subgroups in their profinite completions.

The first homology group with twisted coefficients for the mapping class group of a non-orientable surface of genus three with two boundary components

The first homology group with twisted coefficients for the mapping class group of a non-orientable surface of genus three with two boundary components

Crossed homomorphisms and low dimensional representations of mapping class groups of surfaces

We continue the study of low dimensional linear representations of mapping class groups of surfaces initiated by Franks–Handel [Proc. Amer. Math. So. 141 (2013), pp. 2951–2962] and Korkmaz [Low-dimensional linear representations of mapping class groups, preprint, arXiv:1104.4816v2 (2011)]. We consider ( 2 g + 1 ) (2g+1) -dimensional complex linear representations of the pure mapping class groups of compact orientable surfaces of genus g g . We give a complete classification of such representations for g ≥ 7 g \geq 7 up to conjugation, in terms of certain twisted 1 1 -cohomology groups of the mapping class groups. A new ingredient is to use the computation of a related twisted 1 1 -cohomology group by Morita [Ann. Inst. Fourier (Grenoble) 39 (1989), pp. 777–810]. The classification result implies in particular that there are no irreducible linear representations of dimension 2 g + 1 2g+1 for g ≥ 7 g \geq 7 , which marks a contrast with the case g = 2 g=2 .

Mapping class groups of surfaces with noncompact boundary components

Mapping class groups of surfaces with noncompact boundary components

Presentations of Dehn quandles

Quandles are non-associative algebraic structures with defining axioms modelled on the three Reidemeister moves of planar diagrams of links in the 3-space. The paper gives two approaches to write efficient presentations for a large class of quandles, called Dehn quandles, using presentations of their underlying groups. The first approach gives finite presentations for Dehn quandles of a class of Garside groups and Gaussian groups. The second approach is for general Dehn quandles when the centralisers of generators in their underlying groups are known. Several examples including Dehn quandles of spherical Artin groups, surface groups and mapping class groups of orientable surfaces are given to illustrate the results.

Invariant subalgebras of von Neumann algebras arising from negatively curved groups

Using an interplay between geometric methods in group theory and soft von Neuman algebraic techniques we prove that for any icc, acylindrically hyperbolic group Γ its von Neumann algebra L(Γ) satisfies the so-called ISR property: any von Neumann subalgebraN⊆L(Γ)that is normalized by all group elements in Γ is of the formN=L(Σ)for a normal subgroupΣ◁Γ. In particular, this applies to all groups Γ in each of the following classes: all icc (relatively) hyperbolic groups, most mapping class groups of surfaces, all outer automorphisms of free groups with at least three generators, most graph product groups arising from simple graphs without visual splitting, etc. This result answers positively an open question of Amrutam and Jiang from [2].In the second part of the paper we obtain similar results for factors associated with groups that admit nontrivial (quasi)cohomology valued into various natural representations. In particular, we establish the ISR property for all icc, nonamenable groups that have positive first L2-Betti number and contain an infinite amenable subgroup.

Low‐dimensional linear representations of mapping class groups

AbstractLet be a compact orientable surface of genus with marked points in the interior. Franks–Handel (Proc. Amer. Math. Soc. 141 (2013) 2951–2962)  proved that if then the image of a homomorphism from the mapping class group of to is trivial if and is finite cyclic if . The first result is our own proof of this fact. Our second main result shows that for up to conjugation there are only two homomorphisms from to : the trivial homomorphism and the standard symplectic representation. Our last main result shows that the mapping class group has no faithful linear representation in dimensions less than or equal to . We provide many applications of our results, including the finiteness of homomorphisms from mapping class groups of nonorientable surfaces to , the triviality of homomorphisms from the mapping class groups to or to , and homomorphisms between mapping class groups. We also show that if the surface has marked point but no boundary components, then is generated by involutions if and only if and .

Orbit equivalence rigidity of irreducible actions of right-angled Artin groups

Let $G_\Gamma \curvearrowright X$ and $G_\Lambda \curvearrowright Y$ be two free measure-preserving actions of one-ended right-angled Artin groups with trivial center on standard probability spaces. Assume they are irreducible, i.e. every element from a standard generating set acts ergodically. We prove that if the two actions are stably orbit equivalent (or merely stably $W^*$-equivalent), then they are automatically conjugate through a group isomorphism between $G_\Gamma$ and $G_\Lambda$. Through work of Monod and Shalom, we derive a superrigidity statement: if the action $G_\Gamma \curvearrowright X$ is stably orbit equivalent (or merely stably $W^*$-equivalent) to a free, measure-preserving, mildly mixing action of a countable group, then the two actions are virtually conjugate. We also use the works of Popa and Ioana, Popa and Vaes to establish the $W^*$-superrigidity of Bernoulli actions of all infinite conjugacy classes groups having a finite generating set made of infinite-order elements where two consecutive elements commute, and one has a nonamenable centralizer: these include one-ended nonabelian right-angled Artin groups, but also many other Artin groups and most mapping class groups of finite-type surfaces.

Coarse geometry of pure mapping class groups of infinite graphs

We discuss the large-scale geometry of pure mapping class groups of locally finite, infinite graphs, motivated by recent work of Algom-Kfir–Bestvina [1] and the work of Mann–Rafi [19] on the large-scale geometry of mapping class groups of infinite-type surfaces. Using the framework of Rosendal [24] for coarse geometry of non-locally compact groups, we classify when the pure mapping class group of a locally finite, infinite graph is globally coarsely bounded (an analog of compact) and when it is locally coarsely bounded (an analog of locally compact).Our techniques give lower bounds on the first integral cohomology of the pure mapping class group for some graphs and allow us to compute the asymptotic dimension of all locally coarsely bounded pure mapping class groups of infinite rank graphs. This dimension is always either zero or infinite.

Big Mapping Class Groups and the Co-Hopfian Property

We study injective homomorphisms between big mapping class groups of infinite-type surfaces. First, we construct (uncountably many) examples of surfaces without boundary whose (pure) mapping class groups are not co-Hopfian; these are first such examples of injective endomorphisms of mapping class groups that fail to be surjective. We then prove that, subject to some topological conditions on the domain surface, any continuous injective homomorphism between (arbitrary) big mapping class groups that sends Dehn twists to Dehn twists is induced by a subsurface embedding. Finally, we explore the extent to which, in stark contrast to the finite-type case, superinjective maps between curve graphs impose no topological restrictions on the underlying surfaces.

Small exotic 4-manifolds and symplectic Calabi–Yau surfaces via genus-3 pencils

We explicitly produce symplectic genus-3 Lefschetz pencils (with base points), whose total spaces are homeomorphic but not diffeomorphic to rational surfaces CP^2 # p (-CP^2) for p= 7, 8, 9. We then give a new construction of an infinite family of symplectic Calabi-Yau surfaces with first Betti number b_1=2,3, along with a surface with b_1=4 homeomorphic to the 4-torus. These are presented as the total spaces of symplectic genus-3 Lefschetz pencils we construct via new positive factorizations in the mapping class group of a genus-3 surface. Our techniques in addition allow us to answer in the negative a question of Korkmaz regarding the upper bound on b_1 of a genus-g fibration.

Mapping class group representations from Turaev-Viro TQFT for abelian groups

It is well known that (2 + 1) topological quantum field theories (TQFTs) induce representations for the mapping class groups (MCGs) of surfaces. In this paper, we work explicitly on the MCG representations of any genus from the Turaev-Viro TQFTs based on the category for the finite abelian group G and cocycle . As an application, we determine for torus the image of the MCG representation with the graphical calculus on the string-net diagram.

Asymptotically rigid mapping class groups, I : Finiteness properties of braided Thompson’s and Houghton’s groups

This article is dedicated to the study of asymptotically rigid mapping class groups of infinitely-punctured surfaces obtained by thickening planar trees. Such groups include the braided Ptolemy-Thompson groups $T^\sharp,T^\ast$ introduced by Funar and Kapoudjian, and the braided Houghton groups $\mathrm{br}H_n$ introduced by Degenhardt. We present an elementary construction of a contractible cube complex, on which these groups act with cube-stabilisers isomorphic to finite extensions of braid groups. As an application, we prove Funar-Kapoudjian's and Degenhardt's conjectures by showing that $T^\sharp,T^\ast$ are of type $F_\infty$ and that $\mathrm{br}H_n$ is of type $F_{n-1}$ but not of type $F_n$.

Stable commutator length on big mapping class groups

We study stable commutator length on mapping class groups of certain infinite-type surfaces. In particular, we show that stable commutator length defines a continuous function on the commutator subgroups of such infinite-type mapping class groups. We furthermore show that the commutator subgroups are open and closed subgroups and that the abelianizations are finitely generated in many cases. Our results apply to many popular infinite-type surfaces with locally coarsely bounded mapping class groups.

Torsion normal generators of the mapping class group of a nonorientable surface

We show that the normal closure of any periodic element of the mapping class group of a nonorientable surface whose order is greater than 2 contains the commutator subgroup , which for g ≥ 7 is equal to the twist subgroup, and provide necessary and sufficient conditions for the normal closures of involutions to contain the twist subgroup. Finally, we provide a criterion for a periodic element to normally generate M ( N g ) and give examples.

Generating the mapping class group of a nonorientable surface by three torsions

We prove that the mapping class group $$\mathcal {M}(N_g)$$ of a closed nonorientable surface of genus g different than 4 is generated by three torsion elements. Moreover, for every even integer $$k\ge 12$$ and g of the form $$g=pk+2q(k-1)$$ or $$g=pk+2q(k-1)+1$$ , where p, q are non-negative integers and p is odd, $$\mathcal {M}(N_g)$$ is generated by three conjugate elements of order k. Analogous results are proved for the subgroup of $$\mathcal {M}(N_g)$$ generated by Dehn twists.

Generating the Mapping Class Group of a Nonorientable Surface by Two Elements or by Three Involutions

We prove that, for $$g\ge 19$$ the mapping class group of a nonorientable surface of genus g, $$\mathrm{Mod}(N_g)$$ , can be generated by two elements, one of which is of order g. We also prove that for $$g\ge 26$$ , $$\mathrm{Mod}(N_g)$$ can be generated by three involutions.