The Zieschang–McCool method for generating algebraic mapping-class groups

We prove that, aside from the obvious exceptions, the mapping class group of a compact orientable surface is not abstractly commensurable with any right-angled Artin group. Our argument applies to various sub- groups of the mapping class group—the subgroups generated by powers of Dehn twists and the terms of the Johnson filtration—and additionally to the outer automorphism group of a free group and to certain linear groups. There are many analogies and interconnections between the theories of right- angled Artin groups on one hand and mapping class groups on the other hand. For instance, by the work of Crisp and Wiest (13), the work of Koberda (27), and the work of the first two authors with Mangahas (12), there is an abundance of injective homomorphisms from right-angled Artin groups to mapping class groups. Also, the last two authors proved (30) that any two elements of the pure braid group either generate a free group or a free abelian group—a property shared by all right-angled Artin groups (4, Theorem 1.2). We are thus led to ask to what extent mapping class groups are the same as right-angled Artin groups. It is straightforward to see that most mapping class groups are not isomorphic to right-angled Artin groups, for instance because right-angled Artin groups are torsion free. On the other hand, mapping class groups have finite-index sub- groups that are torsion free, and so this leaves open the possibility that mapping class groups are abstractly commensurable to right-angled Artin groups, that is, that they have isomorphic finite-index subgroups. We prove that, aside from a small number of exceptions, this is not the case. We also extend this result to several classes of groups related to mapping class groups. We start by recalling some definitions. To a finite graph , we can associate a right-angled Artin grou p: this is the group with one generator for each vertex of , and one defin ing relator for each edge, namely, the commutator of the two generators corresponding to the endpoints. Let Sg,n denote a closed, connected, orientable surface of genus g with n marked points. The mapping class group Mod(Sg,n) is the group of homotopy classes of orientation-preserving homeomorphisms of Sg,n preserving the set of marked points. As discussed in Koberda's paper (27, Theorem 1.5), no finite-index subgroup of Mod(Sg,n) injects into a right-angled Artin group if g ≥ 2 and (g,n) 6 (2,0); see

I. Cohomological, combinatorial and algebraic structure: Four questions about mapping class groups by M. Bestvina Some problems on mapping class groups and moduli space by B. Farb Finiteness and Torelli spaces by R. Hain Fifteen problems about the mapping class groups by N. V. Ivanov Problems on homomorphisms of mapping class groups by M. Korkmaz The mapping class group and homotopy theory by I. Madsen Probing mapping class groups using arcs by R. C. Penner Relations in the mapping class group by B. Wajnryb II. Connections with 3-manifolds, symplectic geometry and algebraic geometry: Mapping class group factorizations and symplectic 4-manifolds: Some open problems by D. Auroux The topology of 3-manifolds, Heegaard distances and the mapping class group of a 2-manifold by J. S. Birman Lefschetz pencils and mapping class groups by S. K. Donaldson Open problems in Grothendieck-Teichmuller theory by P. Lochak and L. Schneps III. Geometry and dynamical aspects: Mapping class group dynamics on surface group representations by W. M. Goldman Geometric properties of the mapping class group by U. Hamenstadt Problems on billiards, flat surfaces and translation surfaces by P. Hubert, H. Masur, T. Schmidt, and A. Zorich Problems in the geometry of surface group extensions by L. Mosher Surface subgroups of mapping class groups by A. W. Reid Weil-Petersson perspectives by S. A. Wolpert IV. Braid groups, Out$(F_n)$ and other related groups: Braid groups and Iwahori-Hecke algebras by S. Bigelow Automorphism groups of free groups, surface groups and free abelian groups by M. R. Bridson and K. Vogtmann Problems: Braid groups, homotopy, cohomology, and representations by F. R. Cohen Cohomological structure of the mapping class group and beyond by S. Morita From braid groups to mapping class groups by L. Paris.

We prove that if a normal subgroup of the extended mapping class group of a closed surface has an element of sufficiently small support, then its automorphism group and abstract commensurator group are both isomorphic to the extended mapping class group. The proof relies on another theorem we prove, which states that many simplicial complexes associated to a closed surface have automorphism group isomorphic to the extended mapping class group. These results resolve the metaconjecture of N. V. Ivanov, which asserts that any “sufficiently rich” object associated to a surface has automorphism group isomorphic to the extended mapping class group, for a broad class of such objects. As applications, we show: (1) right-angled Artin groups and surface groups cannot be isomorphic to normal subgroups of mapping class groups containing elements of small support, (2) normal subgroups of distinct mapping class groups cannot be isomorphic if they both have elements of small support, and (3) distinct normal subgroups of the mapping class group with elements of small support are not isomorphic. Our results also suggest a new framework for the classification of normal subgroups of the mapping class group.

Ivanov of the Leningrad Branch of the Steklov Mathematical Institute has shown that the outer automorphism groups of surface mapping class groups are finite. In this report, we shall give explicit descriptions of the outer automorphism groups of both the mapping class groups and extended mapping class groups of closed, connected, orientable surfaces of negative Euler characteristic. (The extended mapping class groups are the extensions of the mapping class groups by the orientation reversing isotopy classes.) For surfaces of genus greater than two, these are, respectively, a cyclic group of order two and the trivial group. For a surface of genus two, these are both noncyclic groups of order four. (Once again, the hypergeometric involution in genus two plays a unique role.)

The group of 2-by-2 matrices with integer entries and determinant $\pm > 1$ can be identified either with the group of outer automorphisms of a rank two free group or with the group of isotopy classes of homeomorphisms of a 2-dimensional torus. Thus this group is the beginning of three natural sequences of groups, namely the general linear groups ${\rm{GL}}(n,\Z)$, the groups of outer automorphisms of free groups of rank $n\geq 2$, and the mapping class groups of closed orientable surfaces of genus $g\geq 1$. Much of the work on mapping class groups and automorphisms of free groups is motivated by the idea that these sequences of groups are strongly analogous. In this article we highlight a few of the most striking similarities and differences between these series of groups and present a list of open problems motivated by this philosophy.

For each integer g≥2 we give the complete list of groups acting as a group of dianalytic automorphisms of a real projective plane with g holes, which, in algebraic terms, correspond to birational automorphisms of real algebraic (M−1)-curves. We also determine those acting as the full group of automorphisms of such a surface. Furthermore, the conjugacy classes of the finite subgroups of its mapping class group are calculated.

We show first that certain automorphism groups of algebraic varieties, and even schemes, are residually finite and virtually torsion free. (A group virtually has a property if some subgroup of finite index has it.) The rest of the paper is devoted to a study of the groups of automorphisms. Aut(Γ) and outer automorphisms Out(Γ) of a finitely generated group Γ, by using the finite-dimensional representations of Γ. This is an old idea (cf. the discussion of Magnus in [11]). In particular the classes of semi-simplen-dimensional representations of Γ are parametrized by an algebraic varietySn(Γ) on which Out(Γ) acts. We can apply the above results to this action and sometimes conclude that Out(Γ) is residually finite and virtually torsion free. This is true, for example, when Γ is a free group, or a surface group. In the latter case Out(Γ) is a “mapping class group.”

We show that any isomorphism between mapping class groups of orientable infinite-type surfaces is induced by a homeomorphism between the surfaces. Our argument additionally applies to automorphisms between finite-index subgroups of these “big” mapping class groups and shows that each finite-index subgroup has finite outer automorphism group. As a key ingredient, we prove that all simplicial automorphisms between curve complexes of infinite-type orientable surfaces are induced by homeomorphisms.

McCarthy's Theorem for the mapping class group of a closed hyperbolic surface states that for any two mapping classes $\sigma,\tau \in \mathrm{Mod}(S)$ there is some power $N$ such that the group $\langle \sigma^N,\tau^N\rangle$ is either free of rank two or abelian, and gives a geometric criterion for the dichotomy. The analogous statement is false in linear groups, and unresolved for outer automorphisms of a free group. Several analogs are known for exponentially growing outer automorphisms satisfying various technical hypothesis. In this article we prove an analogous statement when $\sigma$ and $\tau$ are linearly growing outer automorphisms of $F_r$, and give a geometric criterion for the dichotomy. Further, Hamidi-Tehrani proved that for Dehn twists in the mapping class group this independence dichotomy is \emph{uniform}: $N=4$ suffices. In a similar style, we obtain an $N$ that depends only on the rank of the free group.

An orientation reversing involution \(\sigma\) of a topological compact genus \(g,\, g>2,\) surface \(\Sigma\) induces an antiholomorphic involution \(\sigma^*: T^g \to T^g\) of the Teichmuller space of genus g Riemann surfaces. Two such involutions \(\sigma^*\) and \(\tau^*\) are conjugate in the mapping class group if and only if the corresponding orientation reversing involutions \(\sigma\) and \(\tau\) of \(\Sigma\) are conjugate in the automorphism group of \(\Sigma\). This is equivalent to saying that the quotient surfaces \(\Sigma/\langle \sigma\rangle\) and \(\Sigma/\langle \tau\rangle\) are homeomorphic. Hence the Teichmuller space \(T^g\) has \(m_g = \lfloor{3g+4\over 2}\rfloor\) distinct antiholomorphic involutions, which are also called real structures of \(T^g\) ([7]). This result is a simple fact that follows from Royden's theorem ([4]) stating that the the mapping class group is the full group of holomorphic automorphisms of the Teichmuller space (\(g>2\)). Let \(\sigma^*: T^g\to T^g\) and \(\tau^*: T^g\to T^g\) be two real structures that are not conjugate in the mapping class group. In this paper we construct a real analytic diffeomorphism \(d: T^g\to T^g\) such that

Presentations of mapping class groups and an application to cluster algebras from surfaces

We describe the action of the automorphism group of the complex cubic x + y + x − xyz − 2 on the homology of its fibers. This action includes the action of the mapping class group of a punctured torus on the subvarieties of its SL(2, C) character variety given by fixing the trace of the peripheral element (socalled “relative character varieties”). This mapping class group is isomorphic to PGL(2, Z). We also describe the corresponding mapping class group action for the four-holed sphere and its relative SL(2, C) character varieties, which are fibers of deformations x + y + z − xyz − 2 − Px − Qy − Rz of the above cubic. The 2-congruence subgroup PGL(2, Z)(2) still acts on these cubics and is the full automorphism group when P,Q,R are distinct.

We prove that the mapping class group of a closed connected orientable surface of genus \(g\ge 6\) is generated by two elements of order g. Moreover, for \(g\ge 7\), we obtain a generating set of two elements, of order g and \(g'\), where \(g'\) is the least divisor of g greater than 2. We also prove that the mapping class group is generated by two elements of order \(g/\gcd (g,k)\) for \(g\ge 3k^2+4k+1\) and any positive integer k.

Automorphisms of affine algebraic groups

In this work, we define an orthogonal graph on the set of equivalence classes of tuples over where n and ν are positive integers and or 2. We classify our graph if it is strongly regular or quasi-strongly regular and compute all parameters precisely. We show that our graph is arc transitive. The automorphisms group is given and the chromatic number of the graph except when and ν is odd is determined. Moreover, we work on subconstituents of this orthogonal graph.