Academic literature on the topic 'Automorphisms'

In this dissertation, we describe the automorphism groups for the strong Bruhat orders A_n-1, B_n, and D_n. In particular, the automorphism group of A_n-1 for n ≥ 3 is isomorphic to the dihedral group of order eight, D_4; the automorphism group of B_n for n ≥ 3 is isomorphic to C_2 x C_2 where C_2 is the cyclic group of order two; the automorphism group of D_n for n > 5 and n even is isomorphic to C_2 x C_2 x C_2; and the automorphism group of D_n for n ≥ 5 and n odd is isomorphic to the dihedral group D_4.

This essay is a detailed survey of an article from 1996 published by Franc Forstneric, where he studies symplectic automorphisms of C2n. The vision is to introduce the density property for holomorphic symplectic manifolds. Our idea is that of Dror Varolin when he in 2001 introduced the concept of density property for Stein manifolds. The main result here is the introduction of symplectic shears on C2n equipped with a holomorphic symplectic form and to show that the group generated by finite compositions of symplectic shears is dense in the group of symplectic automorphisms of C2n in the compact-open topology. We give a complete background of the tools from the theory of ordinary differential equations, smooth manifolds, and complex and symplectic geometry that is needed in order to prove this result.<br>Den här uppsatsen är en detaljerad undersökning av en artikel från 1996 publicerad av Franc Forstneric där han studerar symplektiska automorfismer av C2n. Visionen är att introducera täthetsegenskapen för holomorfa symplektiska mångfalder. Våran idé är som den av Dror Varolin när han 2001 introducerade täthetsegenskapen för Stein mångfalder. Huvudresultatet här är införandet av symplektiska skjuvningar på C2n med en holomorfisk symplektisk form och att visa att gruppen som genereras av ändliga sammansättningar av symplektiska skjuvningar är tät i gruppen av symplektiska automorfismer av C2n i den kompakt-öppna topologin. Vi ger en fullständig bakgrund av de verktyg från teorin om ordinära differentialekvationer, släta mångfalder och komplex och symplektisk geometri som behövs för att visa detta.

CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICO<br>Automorfismos genéricos de cubos com alças (handlebodies) aparecem do estudo de classes the isotopia de automorfismos de variedades orientáveis de dimensão três. Automorfismos genéricos permanecem como uma das partes menos entendidas desse estudo.Dado um automorfismo genérico de um cubo com alças, é conhecida uma forma de se construir uma laminação bidimensional que é invariante pelo automorfismo. A essa laminação se associa um fator de crescimento. É sabido que, no caso de tal fator de crescimento ser minimal - uma característica importante, pois mede a complexidade essencial do automorfismo - a laminação deve gozar de uma certa propriedade de incompressibilidade. Nessa tese mostramos que o processo de se achar uma laminação com tal propriedade é algoritmico. Por outro lado, mostramos que tal propriedade não garante que o respectivo fator de crescimento seja minimal. Propomos uma outra propriedade, tensão transversal, mais forte que incompressibilidade, que conjecturamos também ser condição necessária para que o fator de crescimento seja minimal. Provamos a conjectura em alguns casos.Além dos resultados mencionados acima, desenvolvemos métodos para gerar automorfismos genéricos de cubos com alcas, que usamos para apresentar alguma variedade de exemplos.<br>Generic automorphisms of handlebodies appear naturally in the study of isotopy classes of automophisms of orientable three-dimensional manifolds. Generic automorphisms remain as one of the least understood parts of this study. Given a generic automorphism of a handlebody one can construct a bidimensional lamination that is invariant under the automorphism. There is a growth rate associated to this lamination. It is known that, when this growth rate is minimal among all possible choices (an important property, for it measures the essential complexity of the automorphism), the lamination must have a certain incompressibility property. On this thesis we show that the process of

We study automorphisms of irreducible holomorphic symplectic (IHS) manifolds deformation equivalent to the O’Grady’s sixfold. We classify non-symplectic and symplectic automorphisms using lattice theoretic criterions related to the lattice structure of the second integral cohomology. Moreover we introduce the concept of induced automorphisms. There are two birational models for O'Grady's sixfolds, the first one introduced by O'Grady, which is the resolution of singularities of the Albanese fiber of a moduli space of sheaves on an abelian surface, the second one which concerns in the quotient of an Hilbert cube by a symplectic involution. We find criterions to know when an automorphism is induced with respect to these two different models, i.e. it comes from an automorphism of the abelian surface or of the Hilbert cube.

In my thesis I worked on two different projects, both related with projective surfaces with automorphisms. In the first one I studied Abelian surfaces with an automorphism and quaternionic multiplication: this work has already been accepted for publication in the Canadian Journal of Mathematics. In the second project I treat surfaces isogenous to a product of curves and their cohomology. Abelian Surfaces with an Automorphism The Abelian surfaces, with a polarization of a fixed type, whose endomorphism ring is an order in a quaternion algebra are parametrized by a curve, called Shimura curve, in the moduli space of polarized Abelian surfaces. There have been several attempts to find concrete examples of such Shimura curves and of the Abelian surfaces over this curve. In [HM95] Hashimoto and Murabayashi find Shimura curves as the intersection, in the moduli space of principally polarized Abelian surfaces, of Humbert surfaces. Such Humbert surfaces are now known “explicitly” in many other cases and this might allow one to find explicit models of other Shimura curves. Other approaches are taken in [Elk08] and [PS11]. We consider the rather special case where one of the Abelian surfaces in the family is the selfproduct of an elliptic curve. We assume this elliptic curve to have an automorphism of order three or four. For a fixed product polarization of type (1, d), we denote by Hj,d the set of the deformations of the selfproduct with the automorphism of order j. We prove the following theorem: Theorem. Let j ∈ {3,4}, d ∈ Z, d > 0 and let τ ∈ Hj,d, so that the Abelian surface Aτ,d has an automorphism φj of order j. Then the endomorphism algebra of Aτ,d also contains an element ψj with ψj^2 = d. Moreover, for a general τ ∈ Hj,d one has End(Aτ,d)Q =(−j,d)/Q where (a, b)/Q := Q1 ⊕ Qi ⊕ Qj ⊕ Qk is the quaternion algebra with i^2 = a, j^2 =b and ij=−ji. It is easy compute for which d the quaternion algebra (−j,d)/Q is a skew field: for these d the general Abelian surface in the family Hj,d is simple. In particular this provides examples of simple Abelian surfaces with an automorphism of order three and four. This construction, together with well-known results about automorphisms of Abelian surfaces (see [BL04, Chapter 13]), leads to: Theorem. Let A be a simple Abelian surface and φ ∈ Aut(A) a non-trivial automorphism of finite order. Then ord(φ) ∈ {3, 4, 5, 6, 10}. We focus in particular on the family H3,2 of Abelian surfaces with an automorphism of order three and a polarization of type (1,2). In [Bar87] Barth provides a description of a moduli space M2,4, embedded in P5, of (2, 4)-polarized Abelian surfaces with a level structure. Since the polarized Abelian surfaces we consider have an automorphism of order three, the corresponding points in M2,4 are fixed by an automorphism of order three of P^5. This allows us to explicitly identify the Shimura curve in M2,4 that parametrizes the Abelian surfaces with quaternionic multiplication by the maximal order O6 in the quaternion algebra with discriminant 6. It is embedded as a line in M2,4 ⊂ P5 and the symmetric group S4 acts on this line by changing the level structures. According to Rotger [Rot04], an Abelian surface with endomorphism ring O6 is the Jacobian of a unique genus two curve. We show explicitly how to find such genus two curves, or rather their images in the Kummer surface embedded in P5 with a (2,4)-polarization. These curves were already been considered by Hashimoto and Murabayashi in [HM95]: we give the explicit relation between their description and ours. Moreover we find a Humbert surface in M2,4 that parametrizes Abelian surfaces with Z( 2) in the endomorphism ring. Cohomology of surfaces isogenous to a product Surfaces isogenous to a product of curves provide examples of surfaces of general type with many different geometrical invariants. They have been introduced by Catanese in [Cat00]: Definition. A smooth surface S is said to be isogenous to a product (of curves) if it is isomorphic to a quotient (C×D)/G where C and D are curves of genus at least one and G is a finite group acting freely on C × D. We say that a surface isogenous to a product is of mixed type if there exists a element of G interchanging the two curves; otherwise, if G acts diagonally on the product, we say that the surface is of unmixed type. A surface isogenous to a product is of general type if the genus of both curves, C and D, is greater or equal to 2: in this case we say that the surface is isogenous to a higher product. The cohomology groups of a surface S = (C × D)/G isogenous to a product of unmixed type are determined by the action of the group G on the cohomology groups of the curves. Moreover the action of an automorphism group G on a smooth curve C forces a decomposition of the first cohomology group, as described in [BL04, section 13.6] and in [Roj07]: Proposition (Group algebra decomposition). Let G be a finite group acting on a curve C. Let W1, ..., Wr denote the irreducible rational representations of G and let ni := dimDi (Wi), with Di := EndG(Wi), for i = 1, ..., r. Then there are rational Hodge substructures B1, ..., Br such that H1(C, Q) ≃ n1B1+...+nrBr. From this a decomposition of the cohomology groups of the surface S follows directly. We apply these results to surfaces isogenous to a higher product of unmixed type with χ(OS) = 2 and q(S) = 0: they have been studied and classified by Gleissner in [Gle13]. For these surfaces the Hodge diamond is fixed and in particular the Hodge numbers of the second coho- mology groups are the same as those of an Abelian surface. From Gleissner’s classification we obtain a complete list of the 21 possible groups G. We proved that the second cohomology group of these surfaces can be described explicitly as follows: Theorem. Let S be a surface isogenous to a higher product of unmixed type with χ(OS) = 2, q(S) = 0. Then there exist two elliptic curves E1 and E2 such that H2(S, Q) ∼= H2(E1 × E2, Q) as rational Hodge structures. In general it is not possible to construct these elliptic curves “geometrically” using the action of G. More precisely there are no intermediate coverings πF : C → C/F and πH : D → D/H, F, H ≤ G with C/F = E1 and D/H = E2: we can only prove that such elliptic curves must exist. The proof of the theorem is standard for all but four groups: in these cases we study one by one the corresponding surfaces in order to construct the elliptic curves. As a further application we use this approach to study some surfaces isogenous to a higher product with pg = q = 2, in particular those are of Albanese general type.

Let F_n denote the free group of rank n with free basis X. The palindromic automorphism group PiA_n of F_n consists of automorphisms taking each member of X to a palindrome: that is, a word on X that reads the same backwards as forwards. We obtain finite generating sets for certain stabiliser subgroups of PiA_n. We use these generating sets to find an infinite generating set for the so-called palindromic Torelli group PI_n, the subgroup of PiA_n consisting of palindromic automorphisms inducing the identity on the abelianisation of F_n. Two crucial tools for finding this generating set are a new simplicial complex, the so-called complex of partial pi-bases, on which PiA_n acts, and a Birman exact sequence for PiA_n, which allows us to induct on n. We also obtain a rigidity result for automorphism groups of right-angled Artin groups. Let G be a finite simplicial graph, defining the right-angled Artin group A_G. We show that as A_G ranges over all right-angled Artin groups, the order of Out(Aut(A_G)) does not have a uniform upper bound. This is in contrast with extremal cases when A_G is free or free abelian: in these cases, |Out(Aut(A_G))| < 5. We prove that no uniform upper bound exists in general by placing constraints on the graph G that yield tractable decompositions of Aut(A_G). These decompositions allow us to construct explicit members of Out(Aut(A_G)).

Un automorphisme non-symplectique d'ordre fini n sur une surface X de type K3 est un automorphisme σ ∈ Aut(X) qui satisfait σ*(ω) = λω où λ est une racine primitive n-ième de l'unité et ω est le générateur de H2,0(X). Dans cette thèse on s’intéresse aux automorphismes non-symplectiques d'ordre 8 et 16 sur les surfaces K3. Dans un premier temps, nous classifionsles automorphismes non-symplectiques σ d'ordre 8 quand le lieu fixe de sa quatrième puissance σ⁴ contient une courbe de genre positif, on montre plus précisément que le genre de la courbe fixée par σ est au plus un. Ensuite nous étudions le cas où le lieu fixe de σ contient au moins une courbe et toutes les courbes fixées par sa quatrième puissance σ⁴ sont rationnelles. Enfin nous étudions le cas où σ et son carré σ² agissent trivialement sur le groupe de Néron-Severi. Nous classifions toutes les possibilités pour le lieu fixe de σ et de son carré σ² dans ces trois cas. Nous obtenons la classification complète pour les automorphismes non-symplectiques d'ordre 8 sur les surfaces K3. Dans la deuxième partie de la thèse, nous classifions les surfaces K3 avec automorphisme non-symplectique d'ordre 16 en toute généralité. Nous montrons que le lieu fixe contient seulement courbes rationnelles et points isolés et nous classifions complètement les sept configurations possibles. Si le groupe de Néron-Severi a rang 6, alors il y a deux possibilités et si son rang est 14, il y a cinq possibilités. En particulier si l'action de l'automorphisme est trivial sur le groupe de Néron-Severi, alors nous montrons que son rang est six. Enfin, nous construisons des exemples qui correspondent à plusieurs cas dans la classification des automorphismes non-symplectiques d'ordre 8 et nous donnons des exemples pour chaque cas dans la classification des automorphismes non-symplectiques d'ordre 16<br>A non-symplectic automorphism of finite order n on a K3 surface X is an automorphism σ ∈ Aut(X) that satisfies σ*(ω) = λω where λ is a primitive n−root of the unity and ω is a generator of H2,0(X). In this thesis we study the non-symplectic automorphisms of order 8 and 16 on K3 surfaces. First we classify the non-symplectic automorphisms σ of order eight when the fixed locus of its fourth power σ⁴ contains a curve of positive genus, we show more precisely that the genus of the fixed curve by σ is at most one. Then we study the case of the fixed locus of σ that contains at least a curve and all the curves fixed by its fourth power σ⁴ are rational. Finally we study the case when σ and its square σ² act trivially on the Néron-Severi group. We classify all the possibilities for the fixed locus of σ and σ² in these three cases. We obtain a complete classifiction for the non-symplectic automorphisms of order 8 on a K3 surfaces.In the second part of the thesis, we classify K3 surfaces with non-symplectic automorphism of order 16 in full generality. We show that the fixed locus contains only rational curves and isolated points and we completely classify the seven possible configurations. If the Néron-Severi group has rank 6, there are two possibilities and if its rank is 14, there are five possibilities. In particular ifthe action of the automorphism is trivial on the Néron-Severi group, then we show that its rank is six.Finally, we construct several examples corresponding to several cases in the classification of the non-symplectic automorphisms of order 8 and we give an example for each case in the classification of the non-symplectic automorphisms of order 16

In this thesis we explore the problem of computing automorphisms of finite groups, eventually focusing on some group product constructions. Roughly speaking, the automorphism group of a group gives the nature of its internal symmetry. In general, determination of the automorphism group requires significant computational effort and it is advantageous to find situations in which this may be reduced. The two main results give descriptions of the automorphism groups of finite direct products and split metacyclic p-groups. Given a direct product G = H x K where H and K have no common direct factor, we give the order and structure of Aut G in terms of Aut H, Aut K and the central homomorphism groups Hom (H, Z(K)) and Hom (K, Z(H)). A similar result is given for the the split metacyclic p-group, in the case where p is odd. Implementations of both of these results are given as functions for the computational algebra system GAP, which we use extensively throughout. An account of the literature and relevant standard results on automorphisms is given. In particular we mention one of the more esoteric constructions, the automorphism tower. This is defined as the series obtained by repeatedly taking the automorphism group of some starting group G₀. There is interest as to whether or not this series terminates, in the sense that some group is reached that is isomorphic to its group of automorphisms. Besides a famous result of Wielandt in 1939, there has not been much further insight gained here. We make use of the technology to construct several examples, demonstrating their complex and varied behaviour. For the main results we introduce a 2 x 2 matrix description for the relevant automorphism groups, where the entries come from the homorphism groups mentioned previously. In the case of the direct product, this is later generalised to an n x n matrix (when we consider groups with any number of direct factors) and the common direct factor restriction is relaxed to the component groups not having a common abelian direct factor. In the case of the split metacyclic p-group, our matrices have entries that are not all homomorphisms, but are similar. We include the code for our GAP impementation of these results, which we show significantly expedites computation of the automorphism groups. We show that this matrix language can be used to describe automorphisms of any semidirect product and certain central products too, although these general cases are much more complicated. Specifically, multiplication is no longer defined in such a natural way as is seen in the previous cases and the matrix entries are mappings much less well-behaved than homomorphisms. We conclude with some suggestion of types of semidirect products for which our approach may yield a convenient description of the automorphisms.