Real elements in the mapping class group of [formula omitted

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

On loxodromic actions of Artin–Tits groups

We provide a simple criterion for an element of the mapping class group of a closed surface to be a normal generator for the mapping class group. We apply this to show that every nontrivial periodic mapping class that is not a hyperelliptic involution is a normal generator for the mapping class group when the genus is at least 3. We also give many examples of pseudo-Anosov normal generators, answering a question of D. D. Long. In fact we show that every pseudo-Anosov mapping class with stretch factor less than \sqrt{2} is a normal generator. Even more, we give pseudo-Anosov normal generators with arbitrarily large stretch factors and arbitrarily large translation lengths on the curve graph, disproving a conjecture of Ivanov.

Given any family of cubic fields defined by local conditions at finitely many primes, we determine the mean number of 2-torsion elements in the class groups and narrow class groups of these cubic fields when ordered by their absolute discriminants. For an order $\cal O$ in a cubic field, we study the three groups: $\rm Cl_2(\cal O)$, the group of ideal classes of $\cal O$ of order 2; $\rm Cl^+_2(\cal O)$, the group of narrow ideal classes of $\cal O$ of order 2; and ${\cal I}_2(\cal O)$, the group of ideals of $\cal O$ of order 2. We prove that the mean value of the difference $|\rm Cl_2({\cal O})|-\frac14|{\cal I}_2(\cal O)|$ is always equal to $1$, whether one averages over the maximal orders in real cubic fields, over all orders in real cubic fields, or indeed over any family of real cubic orders defined by local conditions. For the narrow class group, we prove that the mean value of the difference $|\rm Cl^+_2({\cal O})|-|{\cal I}_2(\cal O)|$ is equal to $1$ for any such family. For any family of complex cubic orders defined by local conditions, we prove similarly that the mean value of the difference $|\rm Cl_2(\mathcal O)|-\frac12|{\cal I}_2(\cal O)|$ is always equal to $1$, independent of the family. The determination of these mean numbers allows us to prove a number of further results as by-products. Most notably, we prove---in stark contrast to the case of quadratic fields---that: 1) a positive proportion of cubic fields have odd class number; 2) a positive proportion of real cubic fields have isomorphic 2-torsion in the class group and the narrow class group; and 3) a positive proportion of real cubic fields contain units of mixed real signature. We also show that a positive proportion of real cubic fields have narrow class group strictly larger than the class group, and thus a positive proportion of real cubic fields do not possess units of every possible real signature.

1. In studying the structure of abstract groups, one is often confronted with the seemingly trivial question: when do we consider a group (or class of groups) as known? The question has many aspects, and is usually a very complex one. Of course, the group is formally known as soon as its multiplication table is determined; but in most cases, little further information can be obtained from the multiplication table itself on the structural elements of the group, such as subgroups, normal subgroups, automorphisms, etc. If one is however, interested in the classification of groups belonging to a certain class, then the problem is concentrated on the enumeration of groups having some common property, and the structural elements of the individual groups are of a secondary importance. In this case, the problem can be formulated in the following way: Find some finite algebraical procedure, depending on some integral or more general algebraical parameters, which enables us to construct, in an abstract or symbolic way, each group belonging to the class under consideration. A group is of course constructed if its elements are known, and the groupproduct of any two elements can be calculated by means of the defining procedure. It is desirable that the parameters, called invariants of the group, should be characteristic of the group in the sense that no two distinct systems of operations should define the same group. A classical example for such an interpretation of the structure problem is furnished by the fundamental theorem of finite Abelian groups. In this paper we are concerned with the enumeration of certain finite metabelian groups. (Metabelian is a group with Abelian commutator subgroup.) The problem in all its generality is a very difficult one, and far from being solved in the above sense. However, special classes of metabelian groups are accessible to investigation, and such investigations have an interest in so far as they clear more aspects of the general problem. In studying the structure of metabelian groups, the method of group extensions is often found to be the most natural tool. As an extensive use will be made of this method in this paper, I want to formulate it in such a way as it seems most suitable for the case discussed here. A similar but formally different formulation was given by A. Scholz1. See also W. Magnus2. Let ( be a finite metabelian group, additively written, 2t an Abelian normal subgroup of (M such that the quotient group e = (5/Wt is Abelian. Elements

This chapter deals with Dehn twists, the simplest infinite-order elements of Mod(S). It first defines Dehn twists and proves that they are nontrivial elements of the mapping class group. In particular, it considers the action of Dehn twists on simple closed curves. As one application of this study, the chapter proves that if two simple closed curves in Sɡ have geometric intersection number greater than 1, then the associated Dehn twists generate a free group of rank 2 in Mod(S). It also proves some fundamental facts about Dehn twists and describes the center of the mapping class group, along with algebraic relations that can occur between two Dehn twists. Finally, it explores three geometric operations on a surface that each induces an algebraic operation on the corresponding mapping class group: the inclusion homomorphism, the capping homomorphism, and the cutting homomorphism.

We give elementary applications of quasi-homomorphisms to growth problems in groups. A particular case concerns the number of torsion elements required to factor a given element in the mapping class group of a surface.

Let k be an algebraic number field of finite degree or an algebraic function field of dimension 1 with finite constant field and let K/k be a finite, separably Galois extension field. We call an algebraic extension field L, not necessarily Galois nor necessarily of finite degree, of K an everywhere locally abelian extension, denoted in the following as EL-abelian extension, for brevity, of K over k, if and only if for every finite extension L' of K in L and every valuation v of L it holds that the completion field Lv' of L' with reference to v is a composition field of KV and an abelian extension Mv of kv in Lv', where we denote by kv and KV the closures of k and K in Lv', respectively. Let Q be an algebraic closure of K, BK/k the maximum separably EL-abelian extension of K/k in Q, CK/k the maximum separably central extension of K/k in Q (i.e. the maximum extension CK/k which is separably normal over k, contains K, and has the Galois group of CK/k/K in the center of the Galois group of CK/k/k), Ik and IK the groups of the ideles of k and K, Ak and AK the maximum separably abelian extensions of k and K in Q, respectively. Let DK/k=BK/k nCKIk. Then, the norm residue symbol 0Jk in Ak/k is, if restricted into NK/kIK, extensible into a homomorphism UK/k of NK/kIK into the Galois group G(DK/k/K). We shall study in the present article arithmetical meanings of the extensibility. In ?1, we shall define oUK/k precisely, in ?2, study the kernel and obtain a principal genus theorem (Theorem 2), from which will follow easily a certain generalized formulation of some fundamental theorems in the class field theory.1 Combining it with a known property2 of the quotient group of the connected component of the unit element of the idele class group by the natural image of the maximal compact subgroup of the connected component of the unit element of the idele group, we shall obtain, in ?3, our main result (Theorem 8), which concerns total norm residues in the principal idele group of k for K/k, and which is, the author thinks, an idele-theoretic reconstruction of a research of Scholz.3

For any element A of the modular group PSL(2,Z), it follows from work of Bavard that scl(A) is greater than or equal to rot(A)/2, where scl denotes stable commutator length and rot denotes the rotation quasimorphism. Sometimes this bound is sharp, and sometimes it is not. We study for which elements A in PSL(2,Z) the rotation quasimorphism is extremal in the sense that scl(A)=rot(A)/2. First, we explain how to compute stable commutator length in the modular group, which allows us to experimentally determine whether the rotation quasimorphism is extremal for a given A. Then we describe some experimental results based on data from these computations. Our main theorem is the following: for any element of the modular group, the product of this element with a sufficiently large power of a parabolic element is an element for which the rotation quasimorphism is extremal. We prove this theorem using a geometric approach. It follows from work of Calegari that the rotation quasimorphism is extremal for a hyperbolic element of the modular group if and only if the corresponding geodesic on the modular surface virtually bounds an immersed surface. We explicitly construct immersed orbifolds in the modular surface, thereby verifying this geometric condition for appropriate geodesics. Our results generalize to the 3-strand braid group and to arbitrary Hecke triangle groups.

Canonical symplectic representations for prime order conjugacy classes of the mapping-class group

On selection rules of molecules having one, two and three CH 3 tops

The sum of distances between every pair of vertices in a graph G is called the Wiener index of G. This graph invariant was initially utilized to predict certain physico-chemical properties of organic compounds. However, the Wiener index of G does not account for any of its symmetries, which are also known to effect these physico-chemical properties. Graovac and Pisanski modified the Wiener index of G to measure the average distance each vertex is displaced under the elements of the symmetry group of G; we call this the Graovac-Pisanski (GP) distance number of G. In this article, we prove that the set of all GP distance numbers of graphs with isomorphic symmetry groups is dense in a half-line. Moreover, for each finite group Γ and each rational number q within this half-line, we present a construction for a graph whose GP distance number is q and whose symmetry group is isomorphic to Γ. This construction results in graphs whose vertex orbits are not connected; we also consider an analogous construction which ensures that all vertex orbits are connected.

We study the structure of finite groups G which act as elation groups on finite generalized quadrangles and contain a full group of symmetries about some line through the base point. Such groups are related to the translation groups of translation transversal designs with parameters depending on those of the quadrangles. Using results on the structure of p-groups which act as translation groups on transversal designs and results on the index of the Hughes subgroups of finite p-groups, we can show how restricted the structure of elation groups of finite generalized quadrangles with symmetries is. One of our main results is that G is necessarily an elementary abelian 2-group, provided that G has even cardinality. In particular, the elation generalized quadrangle coordinatized by G is a translation generalized quadrangle with G as translation group, that is, G contains full groups of symmetries about every line through the base point.

The symmetry group of a system of differential equations is the largest local group of transformations acting on the independent and dependent variables of the system with the property that it transform solutions of the system to other solutions. The main goal of this chapter is to determine a useful, systematic, computational method that will explicitly determine the symmetry group of any given system of differential equations. We restrict our attention to connected local Lie groups of symmetries, leaving aside problems involving discrete symmetries such as reflections, in order to take full advantage of the infinitesimal techniques developed in the preceding chapter. Before pressing on to the case of differential equations, it is vital that we deal adequately with the simpler situation presented by symmetry groups of systems of algebraic equations, and this is done in the first section. Section 2.2 investigates the precise definition of a symmetry group of a system of differential equations, which requires knowledge of how the group elements actually transform the solutions. The corresponding infinitesimal method rests on the important concept of “prolonging” a group action to the spaces of derivatives of the dependent variables represented in the system. The key “prolongation formula” for an infinitesimal generator of a group of transformations, given in Theorem 2.36, then provides the basis for the systematic determination of symmetry groups of differential equations.

We consider families of quartic number fields whose normal closures over \mathbb{Q} have Galois group isomorphic to D_4 , the symmetries of a square. To any such field L , one can associate the Artin conductor of the corresponding 2-dimensional irreducible Galois representation with image D_4 . We determine the asymptotic number of such D_4 -quartic fields ordered by conductor, and compute the leading term explicitly as a mass formula, verifying heuristics of Kedlaya and Wood. Additionally, we are able to impose any local splitting conditions at any finite number of primes (sometimes, at an infinite number of primes), and as a consequence, we also compute the asymptotic number of order-4 elements in class groups and narrow class groups of quadratic fields ordered by discriminant. Traditionally, there have been two approaches to counting quartic fields, using arithmetic invariant theory in combination with geometry-of-number techniques, and applying Kummer theory together with L -function methods. Both of these strategies fail in the case of D_4 -quartic fields ordered by conductor since counting quartic fields containing a quadratic subfield with large discriminant is difficult. However, when ordering by conductor, we utilize additional algebraic structure arising from the outer automorphism of D_4 combined with both approaches mentioned above to obtain exact asymptotics.