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1
Suabedissen, Rolf. "Topologizing group actions." Thesis, University of Oxford, 2006. http://ora.ox.ac.uk/objects/uuid:ca186b79-b4ea-423f-a697-5ed47b4672f9.
Testo completoAbstract (sommario):
This thesis is centered on the following question: Given an abstract group action by an Abelian group G on a set X, when is there a compact Hausdorff topology on X such that the group action is continuous? If such a topology exists, we call the group action compact-realizable. We show that if G is a locally-compact group, a necessary condition for a G-action to be compact-realizable, is that the image of X under the stabilizer map must be a compact subspace of the collection of closed subgroups of G equipped with the co-compact topology. We apply this result to give a complete characterization for the case when G is a compact Abelian group in terms of the existence of continuous compact Hausdorff pre-images of a certain topological space associated with the group action. If G is not compact, we will show that the necessary condition is not sufficient. Together with various examples, we then present a general two-stage method of construction for compact Hausdorff topologies for R-actions. For discrete groups, the necessary condition above turns out to be not very strong. In the case of G = Z<sup>2</sup> we will see that the two cases |X| < c and |X| ≥ c must be treated very differently. We derive necessary conditions for a group action with |X| < c to be compact-realizable by constructing particularly nice open partitions of the space X. We then use symbolic dynamics together with some generic constructions to obtain a partial converse in this case. If |X| ≥ c we give further constructions of compact Hausdorff topologies for which the group action is continuous.
2
Chung, Kin Hoong School of Mathematics UNSW. "Compact Group Actions and Harmonic Analysis." Awarded by:University of New South Wales. School of Mathematics, 2000. http://handle.unsw.edu.au/1959.4/17639.
Testo completoAbstract (sommario):
A large part of the structure of the objects in the theory of Dooley and Wildberger [Funktsional. Anal. I Prilozhen. 27 (1993), no. 1, 25-32] and that of Rouviere [Compositio Math. 73 (1990), no. 3, 241-270] can be described by considering a connected, finite-dimentional symmetric space G/H (as defined by Rouviere), with ???exponential map???, Exp, from L G/L H to G/H, an action, ???: K ??? Aut??(G) (where Aut?? (G) is the projection onto G/H of all the automorphisms of G which leave H invariant), of a Lie group, K, on G/H and the corresponding action, ???# , of K on L G/L H defined by g ??? L (???g), along with a quadruple (s, E, j, E#), where s is a ???# - invariant, open neighbourhood of 0 in L G/L H, E is a test-function subspace of C??? (Exp s), j ?? C??? (s), and E# is a test-function subspace of C??? (s) which contains { j.f Exp: f ?? E }. Of interest is the question: Is the function ???: ?? ??? ????, where ??: f ??? j.f Exp, a local associative algebra homomorphism from F# with multiplication defined via convolution with respect to a function e: s x s ??? C, to F, with the usual convolution for its multiplication (where F is the space of all ??? - invariant distributions of E and F# is the space of all ???# - invariant distributions of E#)? For this system of objects, we can show that, to some extent, the choice of the function j is not critical, for it can be ???absorbed??? into the function e. Also, when K is compact, we can show that ??? ker ?? = { f ?? E : ???k f (???g) dg = 0}. These results turn out to be very useful for calculations on s2 ??? G/H, where G = SO(3) and H??? SO(3) with H ??? SO(2) with ??? : h ??? Lh, as we can use these results to show that there is no quadruple (s, E, j, E#) for SO(3)/H with j analytic in some neighbourhood of 0 such that ??? is a local homomorphism from F# to F. Moreover, we can show that there is more than one solution for the case where s, E and E# are as chosen by Rouviere, if e is does not have to satisfy e(??,??) = e(??,??).
3
Klaus, Michele. "Group actions on homotopy spheres." Thesis, University of British Columbia, 2011. http://hdl.handle.net/2429/35981.
Testo completoAbstract (sommario):
In the first part of the thesis we discuss the rank conjecture of Benson and Carlson. In particular, we prove that if G is a finite p-group of rank 3 and with p odd, or if G is a central extension of abelian p-groups, then there is a free finite G-CW-complex homotopy equivalent to the product of rk(G) spheres; where rk(G) is the rank of G.
We also treat an extension of the rank conjecture to groups of finite virtual cohomological dimension. In this context, for p a fixed odd prime, we show that there is an infinite group L satisfying the two following properties: every finite subgroup G<L is a p-group with rk(G)<3 and for every finite dimensional L-CW-complex homotopy equivalent to a sphere, there is at least one isotropy subgroup H<L with rk(H)=2.
In the second part of the thesis we discuss the study of homotopy G-spheres up to Borel equivalence. In particular, we provide a new approach to the construction of finite homotopy G-spheres up to Borel equivalence, and we apply it to give some new examples for some semi-direct products.
4
Nica, Bogdan. "Group actions on median spaces." Thesis, McGill University, 2004. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=80342.
Testo completoAbstract (sommario):
We investigate a particular type of geometry, namely median metric spaces. This encompasses median graphs, as well as simple combinatorial structures known as spaces with walls.<br>The group-theoretic applications are towards the Kazhdan property and the Haagerup property.
5
Morris, Gary. "Dynamical constraints on group actions." Thesis, University of East Anglia, 1998. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.267775.
Testo completo
6
Helsdon, Mark Andrew. "Group actions on real trees." Thesis, University of Cambridge, 2007. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.613233.
Testo completo
7
Moseley, Daniel, and Daniel Moseley. "Group Actions on Hyperplane Arrangements." Thesis, University of Oregon, 2012. http://hdl.handle.net/1794/12373.
Testo completoAbstract (sommario):
In this dissertation, we will look at two families of algebras with connections to hyperplane arrangements that admit actions of finite groups. One of the fundamental questions to ask is how these decompose into irreducible representations. For the first family of algebras, we will use equivariant cohomology techniques to reduce the computation to an easier one. For the second family, we will use two decompositions over the intersection lattice of the hyperplane arrangement to aid us in computation.
8
Posthuma, Hessel Bouke. "Quantization of Hamiltonian loop group actions." [S.l. : Amsterdam : s.n.] ; Universiteit van Amsterdam [Host], 2003. http://dare.uva.nl/document/70151.
Testo completo
9
Clarkson, James Price. "Group actions on finite homotopy spheres." Thesis, University of British Columbia, 2010. http://hdl.handle.net/2429/27134.
Testo completoAbstract (sommario):
Recently, Grodal and Smith [7}have developed a finite algebraic model to study hG-spaces where G is a finite group. The procedure associates to each G-space X with finite F_p homology a perfect chain complex of functors over the orbit category. When X has the homotopy type of a sphere, this construction is particularly well behaved. The reverse construction, building an hG-space from the algebraic model, generally produces an infinite dimensional space.
In this thesis, we construct a finiteness obstruction for hG-spheres working one prime at a time. We then begin the development of a global finiteness obstruction. When G is the metacyclic group of order pq, we are able to go further and express the global finiteness obstruction in terms of dimension functions. In addition, we relate the work of tom Dieck and Petrie [19] concerning homotopy representations to the newer model of Grodal and Smith, and compute the rank of V_w(G). We conclude with some new examples of finite Σ₃-spheres.
10
Platten, Richard John. "Proper group actions on CW-complexes." Thesis, Queen Mary, University of London, 1999. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.392890.
Testo completo
11
Braven, Charles Frederick. "Cyclic homology, manifolds and group actions." Thesis, University of Exeter, 2002. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.269737.
Testo completo
12
Orem, Hendrik. "Fast Matrix Multiplication via Group Actions." Scholarship @ Claremont, 2009. https://scholarship.claremont.edu/hmc_theses/221.
Testo completoAbstract (sommario):
Recent work has shown that fast matrix multiplication algorithms can be constructed by embedding the two input matrices into a group algebra, applying a generalized discrete Fourier transform, and performing the multiplication in the Fourier basis. Developing an embedding that yields a matrix multiplication algorithm with running time faster than naive matrix multiplication leads to interesting combinatorial problems in group theory. The crux of such an embedding, after a group G has been chosen, lies in finding a triple of subsets of G that satisfy a certain algebraic relation. I show how the process of finding such subsets can in some cases be greatly simplified by considering the action of the group G on an appropriate set X. In particular, I focus on groups acting on regularly branching trees.
13
Belliart, Michel. "Actions de groupes de Lie sur les variétés compactes." Valenciennes, 1995. https://ged.uphf.fr/nuxeo/site/esupversions/9806b24c-e64d-4e28-b75a-6d3de2b5eb3a.
Testo completoAbstract (sommario):
Cette thèse est en deux parties. Dans la première partie, on énonce, justifie et montre partiellement la conjecture suivante : L'action localement libre de codimension 1, au moins deux fois continument différentiable et préservant le volume d'un groupe de Lie non-unimodulaire sur une variété compacte est conjuguée dans sa classe de différentiabilité à une action homogène. On fournit également des exemples de groupes auxquels ce résultat s'applique. Dans la seconde partie, on répond complètement au problème suivant, posé par J. F. Plante en 1986 : A quelles conditions un groupe de Lie connexe donné peut-il agir continument et sans point fixe global sur une surface compacte donnée ? On en déduit une réponse à trois questions posées par Plante dans le même domaine. Ces deux résultats constituent la partie originale de la thèse. Ils sont précédés de deux survols, l'un de la théorie de Lie, l'autre de celle des surfaces, et la seconde partie est suivie d'une copie du dernier chapitre de la thèse de Mostow, ce qui en facilite la lecture.
14
Niederkrüger, Klaus. "Compact Lie group actions on contact manifolds." [S.l. : s.n.], 2005. http://deposit.ddb.de/cgi-bin/dokserv?idn=975890360.
Testo completo
15
Garcia, Armas Mario. "Group actions on curves over arbitrary fields." Thesis, University of British Columbia, 2015. http://hdl.handle.net/2429/52472.
Testo completoAbstract (sommario):
This thesis consists of three parts. The common theme is finite group actions on algebraic curves defined over an arbitrary field k.
In Part I we classify finite group actions on irreducible conic curves defined over k. Equivalently, we classify finite (constant) subgroups of SO(q) up to conjugacy, where q is a nondegenerate quadratic form of rank 3 defined over k. In the case where k is the field of complex numbers, these groups were classified by F. Klein at the end of the 19th century. In recent papers of A. Beauville and X. Faber, this classification is extended to the case where k is arbitrary, but q is split. We further extend their results by classifying finite subgroups of SO(q) for any base field k of characteristic ≠ 2 and any nondegenerate ternary quadratic form q.
In Part II we address the Hyperelliptic Lifting Problem (or HLP): Given a faithful G-action on ℙ¹ defined over k and an exact sequence 1 → μ₂ → Gʹ→ G → 1, determine the conditions for the existence of a hyperelliptic curve C/k endowed with a faithful Gʹ-action that lifts the prescribed G-action on the projective line. Alternatively, this problem may be regarded as the Galois embedding problem given by the surjection Gʹ ↠ G and the G-Galois extension k(ℙ¹)/k(ℙ¹)G. In this thesis, we find a complete solution to the HLP in characteristic 0 for every faithful group action on ℙ¹ and every exact sequence as above.
In Part III we determine whether, given a finite group G and a base field k of characteristic 0, there exists a strongly incompressible G-curve defined over k. Recall that a G-curve is an algebraic curve endowed with the action of a finite group G. A faithful G-curve C is called strongly incompressible if every dominant G-equivariant rational map of C onto a faithful G-variety is birational. We prove that strongly incompressible G-curves exist if G cannot act faithfully on the projective line over k. On the other hand, if G does embed into PGL₂ over k, we show that the existence of strongly incompressible G-curves depends on finer arithmetic properties of k.<br>Science, Faculty of<br>Mathematics, Department of<br>Graduate
16
Yassawi, Reem. "Multiple mixing of local rank group actions." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1998. http://www.collectionscanada.ca/obj/s4/f2/dsk1/tape11/PQDD_0015/NQ44636.pdf.
Testo completo
17
Kutler, Max B. "Group Actions and Divisors on Tropical Curves." Scholarship @ Claremont, 2011. http://scholarship.claremont.edu/hmc_theses/5.
Testo completoAbstract (sommario):
Tropical geometry is algebraic geometry over the tropical semiring, or min-plus algebra. In this thesis, I discuss the basic geometry of plane tropical curves. By introducing the notion of abstract tropical curves, I am able to pass to a more abstract metric-topological setting. In this setting, I discuss divisors on tropical curves. I begin a study of $G$-invariant divisors and divisor classes.
18
Yassawi, Reem Adel. "Multiple mixing of local rank group actions." Thesis, McGill University, 1998. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=35655.
Testo completoAbstract (sommario):
Suppose G is a countable, Abelian group with an element of infinite order and let X be a mixing finite rank (rank r) action of G on a probability space, or let X be a mixing beta-local rank action of G with beta > 1/2. Suppose further that the Folner sequence { Fn } indexing the towers of the beta-local X satisfies a "bounded intersection property": there is a constant p such that each Fn can intersect no more than p disjoint translates of itself. If X is finite rank, we also suppose that the cardinalities of the r Folner sequences Fin ri=1 are comparable. We show that X is mixing of all orders. When G = Z, this extends the results of Kalikow and Ryzhikov to a large class of "funny" rank one transformations. We follow Ryzhikov's joining technique in our proof: the main theorem follows from showing that any pairwise independent joining of k copies of X is necessarily product measure. This method generalizes Ryzhikov's technique.
19
Potter, Rory. "Derived categories of surfaces and group actions." Thesis, University of Sheffield, 2017. http://etheses.whiterose.ac.uk/19643/.
Testo completoAbstract (sommario):
This thesis focuses on two distinct projects on the bounded derived category of coherent sheaves of surfaces and group actions from different directions. The first project studies bielliptic surfaces, which arise as quotients of products of elliptic curves by a finite group acting freely. We prove a structure theorem describing the group of exact autoequivalences of the bounded derived category of coherent sheaves on a bielliptic surface over C. We also list the generators of the group in some cases. The second project studies semi-orthogonal decompositions of the bounded equivariant derived category of a surface S with an effective action of a finite abelian group G. These semi-orthogonal decompositions are constructed by studying the geometry of the quotient stack [S / G]. We produce new examples of semi-orthogonal decompositions of the equivariant derived category of surfaces with a finite abelian group action. We give a new proof of the Derived McKay correspondence in dimension 2. Using this, we construct semi-orthogonal decompositions of the equivariant derived category of C^2 with an effective action of the Dihedral group D_2n. Moreover, we show that these semi-orthogonal decompositions satisfy a conjecture of Polishchuk and Van den Bergh.
20
Joubert, Paul. "Geometric actions of the absolute Galois group." Thesis, Stellenbosch : University of Stellenbosch, 2006. http://hdl.handle.net/10019.1/2508.
Testo completoAbstract (sommario):
Thesis (MSc (Mathematics))--University of Stellenbosch, 2006.<br>This thesis gives an introduction to some of the ideas originating from A. Grothendieck's
1984 manuscript Esquisse d'un programme. Most of these ideas are related to a new
geometric approach to studying the absolute Galois group over the rationals by considering
its action on certain geometric objects such as dessins d'enfants (called stick figures in
this thesis) and the fundamental groups of certain moduli spaces of curves.
I start by defining stick figures and explaining the connection between these innocent
combinatorial objects and the absolute Galois group. I then proceed to give some background
on moduli spaces. This involves describing how Teichmuller spaces and mapping
class groups can be used to address the problem of counting the possible complex structures
on a compact surface. In the last chapter I show how this relates to the absolute
Galois group by giving an explicit description of the action of the absolute Galois group
on the fundamental group of a particularly simple moduli space. I end by showing how
this description was used by Y. Ihara to prove that the absolute Galois group is contained
in the Grothendieck-Teichmuller group.
21
Cotton, Michael R. "Abelian Group Actions and Hypersmooth Equivalence Relations." Thesis, University of North Texas, 2019. https://digital.library.unt.edu/ark:/67531/metadc1505289/.
Testo completoAbstract (sommario):
We show that any Borel action on a standard Borel space of a group which is topologically isomorphic to the sum of a countable abelian group with a countable sum of lines and circles induces an orbit equivalence relation which is hypersmooth. We also show that any Borel action of a second countable locally compact abelian group on a standard Borel space induces an orbit equivalence relation which is essentially hyperfinite, generalizing a result of Gao and Jackson for the countable abelian groups.
22
DUMITRU, ALINA RALUCA. "COMPACT QUANTUM GROUP ACTIONS ON C*-ALGEBRAS." University of Cincinnati / OhioLINK, 2007. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1178900803.
Testo completo
23
Grecianu, Andrei-Paul. "Group actions on non-archimedean hyperbolic spaces." Thesis, McGill University, 2013. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=119392.
Testo completoAbstract (sommario):
We investigate group actions on a metric space that has some sort of hyperbolicity property. Such metric spaces can arise naturally in the study of hyperbolic, or relatively hyperbolic, groups and of systems of equations over them. We will establish a classification of isometries for non-archimedean hyperbolic spaces similar to that of isometries of classical hyperbolic spaces, various quasi-isometric embeddings of non-geodesic hyperbolic spaces into geodesic ones and a criterion for a group acting on a hyperbolic space to be hyperbolic relative to a point stabilizer. We shall then use these results to establish a description of finitely generated torsion-free groups which allow a faithful action by isometries on a non-archimedean hyperbolic space.<br>On étudie les actions de groupe sur un espace métrique qui possède une propriété d'hyperbolicité quelconque. De tels espaces métriques peuvent apparaître naturellement dans le cadre de l'étude des groupes hyperboliques, ou relativement hyperboliques, et de leurs systèmes d'équations. On prouvera une classification des isométries des espaces hyperboliques non-archimédiens similaire à celle des isométries d'espaces hyperboliques classiques, divers plongements quasi-isométriques d'espaces métriques non-géodésiques dans des espaces géodésiques et un critère pour qu'un group agissant sur un espace hyperbolique soit hyperbolique relatif au stabilisateur d'un point. On utilisera ensuite ces résultats pour obtenir une description des groupes sans torsion finiment engendrés qui permettent une action fidèle par isométries sur un espace hyperbolique non-archimédien.
24
Sandfeldt, Sven. "Local Rigidity of Some Lie Group Actions." Thesis, KTH, Matematik (Avd.), 2020. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-272842.
Testo completoAbstract (sommario):
In this paper we study local rigidity of actions of simply connected Lie groups. In particular, we apply the Nash-Moser inverse function theorem to give sufficient conditions for the action of a simply connected Lie group to be locally rigid. Let $G$ be a Lie group, $H < G$ a simply connected subgroup and $\Gamma < G$ a cocompact lattice. We apply the result for general actions of simply connected groups to obtain sufficient conditions for the action of $H$ on $\Gamma\backslash G$ by right translations to be locally rigid. We also discuss some possible applications of this sufficient condition<br>I den här texten så studerar vi lokal rigiditet av gruppverkan av enkelt sammanhängande Liegrupper. Mer specifikt, vi applicerar Nash-Mosers inversa funktionssats för att ge tillräckliga villkor för att en gruppverkan av en enkelt sammanhängande grupp ska vara lokalt rigid. Låt $G$ vara en Lie grupp, $H < G$ en enkelt sammanhängande delgrupp och $\Gamma < G$ ett kokompakt gitter. Vi applicerar resultatet för generella gruppverkan av enkelt sammanhängande grupper för att få tillräckliga villkor för att verkan av $H$ på $\Gamma\backslash G$ med translationer ska vara lokalt rigid. Vi diskuterar också några möjliga tillämpningar av det tillräckliga villkoret.
25
Iverson, Joseph. "Frames Generated by Actions of Locally Compact Groups." Thesis, University of Oregon, 2016. http://hdl.handle.net/1794/20443.
Testo completoAbstract (sommario):
Let $G$ be a second countable, locally compact group which is either compact or abelian, and let $\rho$ be a unitary representation of $G$ on a separable Hilbert space $\mathcal{H}_\rho$. We examine frames of the form $\{ \rho(x) f_j \colon x \in G, j \in I\}$ for families $\{f_j\}_{j \in I}$ in $\mathcal{H}_\rho$. In particular, we give necessary and sufficient conditions for the joint orbit of a family of vectors in $\mathcal{H}_\rho$ to form a continuous frame.
We pay special attention to this problem in the setting of shift invariance. In other words, we fix a larger second countable locally compact group $\Gamma \supset G$ containing $G$ as a closed subgroup, and we let $\rho$ be the action of $G$ on $L^2(\Gamma)$ by left translation. In both the compact and the abelian settings, we introduce notions of Zak transforms on $L^2(\Gamma)$ which simplify the analysis of group frames. Meanwhile, we run a parallel program that uses the Zak transform to classify closed subspaces of $L^2(\Gamma)$ which are invariant under left translation by $G$. The two projects give compatible outcomes.
This dissertation contains previously published material.
26
Kenneally, Darren John. "On eigenvectors for semisimple elements in actions of algebraic groups." Thesis, University of Cambridge, 2010. https://www.repository.cam.ac.uk/handle/1810/224782.
Testo completoAbstract (sommario):
Let G be a simple simply connected algebraic group defined over an algebraically closed field K and V an irreducible module defined over K on which G acts. Let E denote the set of vectors in V which are eigenvectors for some non-central semisimple element of G and some eigenvalue in K*. We prove, with a short list of possible exceptions, that the dimension of Ē is strictly less than the dimension of V provided dim V > dim G + 2 and that there is equality otherwise. In particular, by considering only the eigenvalue 1, it follows that the closure of the union of fixed point spaces of non-central semisimple elements has dimension strictly less than the dimension of V provided dim V > dim G + 2, with a short list of possible exceptions. In the majority of cases we consider modules for which dim V > dim G + 2 where we perform an analysis of weights. In many of these cases we prove that, for any non-central semisimple element and any eigenvalue, the codimension of the eigenspace exceeds dim G. In more difficult cases, when dim V is only slightly larger than dim G + 2, we subdivide the analysis according to the type of the centraliser of the semisimple element. Here we prove for each type a slightly weaker inequality which still suffices to establish the main result. Finally, for the relatively few modules satisfying dim V ≤ dim G + 2, an immediate observation yields the result for dim V < dim B where B is a Borel subgroup of G, while in other cases we argue directly.
27
Wein, Alexander Spence. "Statistical estimation in the presence of group actions." Thesis, Massachusetts Institute of Technology, 2018. http://hdl.handle.net/1721.1/117314.
Testo completoAbstract (sommario):
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2018.<br>Cataloged from PDF version of thesis.<br>Includes bibliographical references (pages 199-210).<br>Imagine we want to recover an unknown vector given many noisy copies of it, except each copy is cyclically shifted by an unknown offset (this is "multi-reference alignment"). Or imagine we want to reconstruct an unknown 3D structure (e.g. a molecule) given many noisy pictures of it taken from different unknown angles (this is "cryo-EM"). These problems (and many others) involve the action of unknown group elements drawn randomly from a compact group such as Z/p or SO(3). In this thesis we study two statistical models for estimation in the presence of group actions. The first is the synchronization model in which we attempt to learn an unknown collection of group elements based on noisy pairwise comparisons. The second is the orbit recovery model in which we observe noisy copies of a hidden signal, each of which is acted upon by a random group element. For both of these models, we explore the fundamental statistical limits as well as the fundamental computational limits (i.e. how well can a polynomial-time algorithm perform?). We use methods from a wide variety of areas, including statistical physics, approximate message passing, representation theory, contiguity and the associated second moment method, invariant theory, algebraic geometry, and the sum-of-squares hierarchy..<br>by Alexander Spence Wein.<br>Ph. D.
28
Jarrett, Kieran. "Non-singular actions of countable groups." Thesis, University of Bath, 2018. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.761021.
Testo completoAbstract (sommario):
In this thesis we study actions of countable groups on measure spaces underthe assumption that the dynamics are non-singular, with particular reference topointwise ergodic theorems and their relationship to the critical dimensions ofthe action.
29
Baumgaertel, H., A. L. Carey, and Andreas Cap@esi ac at. "Hilbert C*-systems for Actions of the Circle Group." ESI preprints, 2000. ftp://ftp.esi.ac.at/pub/Preprints/esi940.ps.
Testo completo
30
Björklund, Michael. "Limit Theorems for Ergodic Group Actions and Random Walks." Doctoral thesis, KTH, Matematik (Inst.), 2009. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-10451.
Testo completoAbstract (sommario):
This thesis consists of an introduction, a summary and 7 papers. The papers are devoted to problems in ergodic theory, equidistribution on compact manifolds and random walks on groups. In Papers A and B, we generalize two classical ergodic theorems for actions of abelian groups. The main result is a generalization of Kingman’s subadditive ergodic theorem to ergodic actions of the group Zd. In Papers C,D and E, we consider equidistribution problems on nilmanifolds. In Paper C we study the asymptotic behavior of dilations of probability measures on nilmanifolds, supported on singular sets, and prove, under some technical assumptions, effective convergences to Haar measure. In Paper D, we give a new geometric proof of an old result by Koksma on almost sure equidistribution of expansive sequences. In paper E we give necessary and sufficient conditions on a probability measure on a homogeneous Riemannian manifold to be non–atomic. Papers F and G are concerned with the asymptotic behavior of random walks on groups. In Paper F we consider homogeneous random walks on Gromov hyperbolic groups and establish a central limit theorem for random walks satisfying some technical moment conditions. Paper G is devoted to certain Bernoulli convolutions and the regularity of their value distributions.<br>QC 20100705
31
Madore, Blair F. "Rank-one group actions with simple mixing Z subactions." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 2000. http://www.collectionscanada.ca/obj/s4/f2/dsk2/ftp03/NQ49929.pdf.
Testo completo
32
Kirsebom, Maxim Solund. "Extreme value theory for group actions on homogeneous spaces." Thesis, University of Bristol, 2014. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.664970.
Testo completoAbstract (sommario):
In this thesis we study extreme value theory for random walks as well as one-parameter actions on homogeneous spaces. In both cases we investigate the limiting distributions for the maximum of an observable evaluated along a trajectory of the system. In particular we are going to consider asymptotic distributions for closest distance returns to a given point· and tor maximal excursions to the cusp. For closest returns on the torus we establish an exact extreme value distribution while for other cases we obtain estimates on the extreme value distributions for sparse sequences. For random walks we also obtain logarithm laws for the maximum. Finally we look into the extreme value statistics of exceedances of high levels in these settings. For the closest returns we establish convergence to a Poisson process for the point process of exceedances. In other cases we obtain estimates on the limiting distribution of the k'th largest maximum for sparse sequences.
33
Björklund, Michael. "Limit theorems for ergodic group actions and random walks /." Stockholm : Matematik, Skolan för teknikvetenskap, Kungliga Tekniska högskolan, 2009. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-10451.
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Birembaux, Olivier. "Actions de groupes résolubles scindement de f-fibres hermitiens." Valenciennes, 1997. https://ged.uphf.fr/nuxeo/site/esupversions/b7bd7233-5e92-4465-8cce-ff9ef4078593.
Testo completoAbstract (sommario):
Première partie : Soient G un groupe de Lie connexe de dimension n - 1, f une action localement libre de classe Cr (avec r supérieure ou égale à 2) de G sur une variété compacte M de dimension n supérieure ou égale a 3. Nous supposons qu'il existe dans l'algèbre de Lie de G un champ Y tel que la partie réelle de chaque valeur propre de ad(Y) soit strictement plus petite que 0. Alors, nous montrons que f est Cr -conjuguée à une action modèle de G sur un espace homogène H/T ou H est un groupe de Lie contenant G. Nos hypothèses impliquent que G a une structure particulière, mais il existe quand même de nombreux exemples : Notamment, la famille des groupes G ayant cette propriété est continue en toute dimension plus grande que 4 ; pour un choix générique de G, le groupe H correspondant n'a aucun quotient compact de dimension n, et ceci fournit une famille continue de groupes de Lie ne possédant aucune action de codimension 1 suffisamment régulière sur une variété compacte. Ces résultats sont liés à la théorie d'Anosov. Deuxième partie : Soient M une variété C connexe, munie d'un flot F et E un F-fibré hermitien au-dessus de M. On donne une version basique du théorème de Leray-Hirsch pour le fibré P(E), projectifié de E. Ensuite on établit un « principe de scindement » feuilleté, i. E on montre qu'il existe une variété B munie d'un flot V et une application feuilletée σ : B vers M telles que le fibré vectoriel image réciproque σ -1 E se décompose en somme directe de V-fibrés hermitiens Si de rang 1 et l'application σ*, induite en cohomologie basique, est injective.
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Cannizzo, Jan. "Schreier Graphs and Ergodic Properties of Boundary Actions." Thesis, Université d'Ottawa / University of Ottawa, 2014. http://hdl.handle.net/10393/31444.
Testo completoAbstract (sommario):
This thesis is broadly concerned with two problems: investigating the ergodic properties of boundary actions, and investigating various properties of Schreier graphs. Our main result concerning the former problem is that, in a variety of situations, the action of an invariant random subgroup of a group G on a boundary of G (e.g. the hyperbolic boundary, or the Poisson boundary) is conservative (there are no wandering sets). This addresses a question asked by Grigorchuk, Kaimanovich, and Nagnibeda and establishes a connection between invariant random subgroups and normal subgroups. We approach the latter problem from a number of directions (in particular, both in the presence and the absence of a probability measure), with an emphasis on what we term Schreier structures (edge-labelings of a given graph which turn it into a Schreier coset graph). One of our main results is that, under mild assumptions, there exists a rich space of invariant Schreier structures over a given unimodular graph structure, in that this space contains uncountably many ergodic measures, many of which we are able to describe explicitly.
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Lange, Carsten. "Combinatorial curvatures, group actions, and colourings: aspects of topological combinatorics." [S.l.] : [s.n.], 2005. http://deposit.ddb.de/cgi-bin/dokserv?idn=973473487.
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Yabu, Yoshiro. "Geometric Study of Classical and Quantum Systems with Group Actions." 京都大学 (Kyoto University), 2009. http://hdl.handle.net/2433/123853.
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SADE, MARTIN CHARLES. "VARIATIONAL PRINCIPLES FOR FIELD VARIABLES SUBJECT TO GROUP ACTIONS (GAUGE)." Diss., The University of Arizona, 1985. http://hdl.handle.net/10150/188075.
Testo completoAbstract (sommario):
This dissertation is concerned with variational problems whose field variables are functions on a product manifold M x G of two manifolds M and G. These field variables transform as type (0,1) tensor fields on M and are denoted by ψ(h)ᵅ (h = 1, ..., n = dim M, α = 1, ..., r = dim G). The dependence of ψ(h)ᵅ on the coordinates of G is given by a generalized gauge transformation that depends on a local map h:M → G. The requirement that a Lagrangian that is defined in terms of these field variables be independent of the coordinates of G and the choice of the map h endows G with a local Lie group structure. The class of Lagrangians that exhibits this type of invariance may be characterized by three invariance identities. These identities, together with an arbitrary solution of a system of partial differential equations, may be used to define field strengths associated with the ψ(h)ᵅ as well as connection and curvature forms on M. The former may be used to express the Euler-Lagrange equations in a particularly simple form. An energy-momentum tensor may also be defined in the usual manner; however additional conditions must be imposed in order to guarantee the existance of conservation laws resulting from this tensor. The above analysis may be repeated for the case that the field variables behave as type (0,2) tensor fields under coordinate transformations on M. For these field variables, the Euler-Lagrange expressions may be expressed as a product of a covariant divergence with the components λʰ of a type (1,0) vector field on M. An unexpected consequence of this construction is the fact that the Euler-Lagrange equations that result for the vector field λʰ are satisfied whenever the Euler-Lagrange equations associated with the field variables are satisfied.
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Wong, Ming Lai. "Q-Fourier transform, q-Heisenberg algebra and quantum group actions /." View Abstract or Full-Text, 2003. http://library.ust.hk/cgi/db/thesis.pl?MATH%202003%20WONG.
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Tarzi, Sam. "Group actions on amorphous sets and reducts of coloured ransom graphs." Thesis, Queen Mary, University of London, 2003. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.406911.
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Alsaody, Seidon. "A Categorical Study of Composition Algebras via Group Actions and Triality." Doctoral thesis, Uppsala universitet, Algebra och geometri, 2015. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-248519.
Testo completoAbstract (sommario):
A composition algebra is a non-zero algebra endowed with a strictly non-degenerate, multiplicative quadratic form. Finite-dimensional composition algebras exist only in dimension 1, 2, 4 and 8 and are in general not associative or unital. Over the real numbers, such algebras are division algebras if and only if they are absolute valued, i.e. equipped with a multiplicative norm. The problem of classifying all absolute valued algebras and, more generally, all composition algebras of finite dimension remains unsolved. In dimension eight, this is related to the triality phenomenon. We approach this problem using a categorical language and tools from representation theory and the theory of algebraic groups. We begin by considering the category of absolute valued algebras of dimension at most four. In Paper I we determine the morphisms of this category completely, and describe their irreducibility and behaviour under the actions of the automorphism groups of the algebras. We then consider the category of eight-dimensional absolute valued algebras, for which we provide a description in Paper II in terms of a group action involving triality. Then we establish general criteria for subcategories of group action groupoids to be full, and applying this to the present setting, we obtain hitherto unstudied subcategories determined by reflections. The reflection approach is further systematized in Paper III, where we obtain a coproduct decomposition of the category of finite-dimensional absolute valued algebras into blocks, for several of which the classification problem does not involve triality. We study these in detail, reducing the problem to that of certain group actions, which we express geometrically. In Paper IV, we use representation theory of Lie algebras to completely classify all finite-dimensional absolute valued algebras having a non-abelian derivation algebra. Introducing the notion of quasi-descriptions, we reduce the problem to the study of actions of rotation groups on products of spheres. We conclude by considering composition algebras over arbitrary fields of characteristic not two in Paper V. We establish an equivalence of categories between the category of eight-dimensional composition algebras with a given quadratic form and a groupoid arising from a group action on certain pairs of outer automorphisms of affine group schemes
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Vigolo, Federico. "Geometry of actions, expanders and warped cones." Thesis, University of Oxford, 2018. http://ora.ox.ac.uk/objects/uuid:0b094203-6f94-4b3b-826e-c8b1ac6203b8.
Testo completoAbstract (sommario):
In this thesis we introduce a notion of graphs approximating actions of finitely generated groups on metric and measure spaces. We systematically investigate expansion properties of said graphs and we prove that a sequence of graphs approximating a fixed action ρ forms a family of expanders if and only if ρ is expanding in measure. This enables us to rely on a number of known results to construct numerous new families of expander (and superexpander) graphs. Proceeding in our investigation, we show that the graphs approximating an action are uniformly quasi-isometric to the level sets of the associated warped cone. The existence of such a relation between approximating graphs and warped cones has twofold advantages: on the one hand it implies that warped cones arising from actions that are expanding in measure coarsely contain families of expanders, on the other hand it provides a geometric model for the approximating graphs allowing us to study the geometry of the expander thus obtained. The rest of the work is devoted to the study of the coarse geometry of warped cones (and approximating graphs). We do so in order to prove rigidity results which allow us to prove that our construction is flexible enough to produce a number of non coarsely equivalent new families of expanders. As a by-product, we also show that some of these expanders enjoy some rather peculiar geometric properties, e.g. we can construct expanders that are coarsely simply connected.
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Ericsson, Sandra, and Marielle Frid. "Suggested actions to improve the warehouse layout at Nordic E-commerce group." Thesis, Linnéuniversitetet, Institutionen för ekonomistyrning och logistik (ELO), 2013. http://urn.kb.se/resolve?urn=urn:nbn:se:lnu:diva-26740.
Testo completoAbstract (sommario):
Background: Nordic E-commerce group (NECG) consists of affiliated companies NordicInk, NordicFeel and PetsOnline. Three different types of companies in one warehouse area contribute to a wide range of products held in inventory. NECG’s problem derives in the fact that the company’s warehouse layout needs to handle a larger volume of sales, but is not capable of doing this in a structured way today. This matter of fact has led to new thoughts concerning improvements of the layout. The warehouse layout referred to in this thesis includes input and output doors, receiving and outgoing areas, material handling equipment and the placement of articles in inventory. Although article placement is per definition a part of warehouse layout, it will be given extra emphasis in this thesis as it is one of NECG’s most critical issues. Purpose: The purpose of this thesis is to identify NECG’s current warehouse layout and article placement and the problems associated with it, as well as to provide suggested actions for improvements. This is to be done so NECG can better utilize the warehouse space, and thus prepare the warehouse to handle a larger volume of sales. Methodology: Qualitative and quantitative data was collected through semi-structured interviews and observations at NECG, as well as through excel files containing transactional data for NordicInk’s products. Through a non-probability, snowball selection the respondents were chosen. There have not been any requirements from NECG regarding confidentiality in terms of the warehouse layout. Findings: There are a number of suggested actions that NECG may apply to improve the warehouse layout; two separate doors for incoming and outgoing deliveries, one designated receiving and one outgoing area, and a flow shop layout combined with a U-form layout would be the most beneficial for NECG’s warehouse. Furthermore, NECG needs a more consistent article placement and it is recommended for the company to apply a floating location system to its high-picked products and a fixed location system for its low-picked products. This will make NECG take better use of the available space and a more structured warehouse will be achieved.<br>Bakgrund: Nordic E-commerce Group (NECG) består utav dotterbolagen NordicInk, NordicFeel och PetsOnline. Då lagret består utav tre olika företag finns ett brett utbud av produkter. NECG:s problem härrör i att företagets lagerlayout inte har följt med i företagets utveckling. För att kunna hantera en större försäljningsvolym, behöver lagerområdet kunna hantera nuvarande och framtida lagervolymer, men är inte kapabel till att göra detta på ett strukturerat sätt i dag. Detta har lett till nya tankar om förbättringsförslag kring lagerlayout. I detta examensarbete avser lagerlayout ingående och utgående portar, inkommande och utgående lagringsområden, materialhanteringsutrustning och artikelplacering i lager. Även om artikeln placering är per definition en del av lagerlayout, kommer det att ges extra fokus i detta examensarbete eftersom det är en av NECG mest kritiska frågor. Syfte: Syftet med detta examensarbete är att identifiera NECG nuvarande lagerlayout och artikel placering samt vilka eventuella problem som finns, för att kunna ge förslag på förbättringar. Detta ska göras så att NECG kan utnyttja sitt lagerutrymme bättre och förbereda sig inför ökade försäljningsvolymer. Metod: Kvalitativ och kvantitativ data samlades in genom semistrukturerade intervjuer och observationer på NECG, samt genom excel filer innehållande transaktionsdata kring NordicInk’s produkter. Genom ett icke-sannolikt, snöbollsurval valdes respondenterna ut. NECG har inte uttryckt några krav om att sekretessbelägga arbetet kring lagerlayout. Resultat: Det finns ett antal förslag på åtgärder som NECG kan tillämpa för att förbättra lagerlayouten: två separata dörrar för inkommande och utgående leveranser, ett designerat område för inkommande och ett för utgående leveranser samt en flow shop layout kombinerat med en U-form layout skulle vara mest fördelaktigt för NECG:s lager. Vidare behöver NECG en mer konsekvent artikelplacering och det rekommenderas för företaget att tillämpa ett flytande system för dess högfrekvent plockade produkter och ett fast system för dess lågfrekvent plockade produkter. Detta kommer att göra så att NECG bättre utnyttjar det tillgängliga lagerutrymmet och ett mer strukturerat system kommer att uppnås. Bakgrund: Nordic E-commerce Group (NECG) består utav dotterbolagen NordicInk, NordicFeel och PetsOnline. Då lagret består utav tre olika företag finns ett brett utbud av produkter. NECG:s problem härrör i att företagets lagerlayout inte har följt med i företagets utveckling. För att kunna hantera en större försäljningsvolym, behöver lagerområdet kunna hantera nuvarande och framtida lagervolymer, men är inte kapabel till att göra detta på ett strukturerat sätt i dag. Detta har lett till nya tankar om förbättringsförslag kring lagerlayout. I detta examensarbete avser lagerlayout ingående och utgående portar, inkommande och utgående lagringsområden, materialhanteringsutrustning och artikelplacering i lager. Även om artikeln placering är per definition en del av lagerlayout, kommer det att ges extra fokus i detta examensarbete eftersom det är en av NECG mest kritiska frågor. Syfte: Syftet med detta examensarbete är att identifiera NECG nuvarande lagerlayout och artikel placering samt vilka eventuella problem som finns, för att kunna ge förslag på förbättringar. Detta ska göras så att NECG kan utnyttja sitt lagerutrymme bättre och förbereda sig inför ökade försäljningsvolymer. Metod: Kvalitativ och kvantitativ data samlades in genom semistrukturerade intervjuer och observationer på NECG, samt genom excel filer innehållande transaktionsdata kring NordicInk’s produkter. Genom ett icke-sannolikt, snöbollsurval valdes respondenterna ut. NECG har inte uttryckt några krav om att sekretessbelägga arbetet kring lagerlayout. Resultat: Det finns ett antal förslag på åtgärder som NECG kan tillämpa för att förbättra lagerlayouten: två separata dörrar för inkommande och utgående leveranser, ett designerat område för inkommande och ett för utgående leveranser samt en flow shop layout kombinerat med en U-form layout skulle vara mest fördelaktigt för NECG:s lager. Vidare behöver NECG en mer konsekvent artikelplacering och det rekommenderas för företaget att tillämpa ett flytande system för dess högfrekvent plockade produkter och ett fast system för dess lågfrekvent plockade produkter. Detta kommer att göra så att NECG bättre utnyttjar det tillgängliga lagerutrymmet och ett mer strukturerat system kommer att uppnås.
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Lenhardt, Fabian [Verfasser]. "The Baum-Connes conjecture, Swan group actions and controlled topology / Fabian Lenhardt." Berlin : Freie Universität Berlin, 2013. http://d-nb.info/1031106510/34.
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Tait, Joseph. "Group actions on differentials of curves and cohomology bases of hyperelliptic curves." Thesis, University of Southampton, 2014. https://eprints.soton.ac.uk/373877/.
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Mazur, Justin D. "Motivic zeta-functions and the Grothendieck ring for varieties with group actions." [Bloomington, Ind.] : Indiana University, 2009. http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3354903.
Testo completoAbstract (sommario):
Thesis (Ph.D.)--Indiana University, Dept. of Mathematics, 2009.<br>Title from PDF t.p. (viewed on Feb. 4, 2010). Source: Dissertation Abstracts International, Volume: 70-04, Section: B, page: 2339. Adviser: Valery Lunts.
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Raizada, Rachana. "Corporate responses to government and environmental group actions designed to protect the environment." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1998. http://www.collectionscanada.ca/obj/s4/f2/dsk2/ftp02/NQ27230.pdf.
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Ash, David B. "Transformational leadership and organizational learning : leader actions that stimulate individual and group learning." Virtual Press, 1997. http://liblink.bsu.edu/uhtbin/catkey/1063425.
Testo completoAbstract (sommario):
The purpose of this research was to identify and describe the action's leaders practice that stimulate individual and group learning. An MLQ- Multifactor Leadership Questionnaire and a series of interviews with members of three work teams were utilized to gather research data. This is one of the first studies completed to research this important aspect of leadership. In all three cases, the teams shared experiences where learning was transformed into useable knowledge to accomplish objectives or solve problems. The three teams learned and generated new knowledge based on viewing, challenging, sharing, orchestrating, and modeling.The findings suggest transformational leadership behaviors and actions can and often do influence individual and group learning. Specific findings included that in all three case studies, 1) team leaders engaged in transformational leadership practices 2) transformational leaders created a climate for learning by encouragement, establishing cooperation, and the identification and use of team talent. In addition, 3) the leader modeled appropriate behaviors, 4) challenged the team to question basic assumptions shared by others, 5) and acquired, shares and assimilated information for the purpose of aiding in team learning. Furthermore, in at least two cases, the leader 6) provided the team with opportunities to learn about the "big picture" and how their actions and decisions impact larger systems, and 7) provided team members with opportunities to become their own leader.Results of the research suggest organizations should consider ways to teach leaders about their role in organizational learning, systems thinking, and how their personal actions influence follower performance. Leaders should also construct climates that allow learning to take place. Lastly, with the support from leaders, knowledge and information systems are important organizational elements, which lead to learning.<br>Department of Educational Leadership
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Ramasu, Pako. "Internal monoid actions in a cartesian closed category and higher-dimensional group automorphisms." Doctoral thesis, University of Cape Town, 2015. http://hdl.handle.net/11427/20248.
Testo completoAbstract (sommario):
The notion of cat¹-group which was introduced by Loday is equivalent to the notions of crossed module and of internal category in the category of groups. This notion of cat¹-groups and their morphisms admits natural generalization to catⁿ-groups, which give rise to n-fold categories in the category of groups. There is also a characterization of catⁿ-groups in terms of crossed n-cubes which was given by Ellis and Steiner. The category Catⁿ (Groups) of internal n-fold categories in the category of groups is a cartesian closed category, however given an object X in Catⁿ (Groups), calculating corresponding action representing object Aut (X) directly would require an enormous calculations. The main purpose of the thesis is to describe that object avoiding such calculations as much as possible. The main tool used in the thesis, apart from the theory of cartesian closed categories, is Loday's theory of catⁿ-groups. We de ne a catⁿ-group X as an additive Mₙ-group X , and then construct the corresponding Aut (X), where Mₙ is a monoid. Since the category of catⁿ-groups is equivalent to Catⁿ (Groups) and since the cartesian closed category Sets Mₙ of Mₙ-sets is much easier to handle than the cartesian closed category of n-fold categories, we shall work just with catⁿ-groups. To assert that, Aut (X) is an action representing object in Sets Mₙ , is to as- sert that, there is a canonical bijection between B-actions of catⁿ-group B on X and the internal group homomorphism B --> Aut (X). Thus, we confirm the construction of Aut (X) by establishing that bijection. Finally, as one of the results of this work, we give the comparison between our cat¹-group Aut (X) and Norrie's actor crossed module (D (G;Z) ;Aut (Z;G;p) w) of a crossed module (Z;G;p) in dimension one.
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Hedén, Isac. "Ga-actions on Complex Affine Threefolds." Doctoral thesis, Uppsala universitet, Matematiska institutionen, 2013. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-203708.
Testo completoAbstract (sommario):
This thesis consists of two papers and a summary. The papers both deal with affine algebraic complex varieties, and in particular such varieties in dimension three that have a non-trivial action of one of the one-dimensional algebraic groups Ga := (C, +) and Gm := (C*, ·). The methods used involve blowing up of subvarieties, the correspondances between Ga - and Gm - actions on an affine variety X with locally nilpotent derivations and Z-gradings respectively on O(X) and passing from a filtered algebra A to its associated graded algebra gr(A). In Paper I, we study Russell’s hypersurface X , i.e. the affine variety in the affine space A4 given by the equation x + x2y + z3 + t2 = 0. We reprove by geometric means Makar-Limanov’s result which states that X is not isomorphic to A3 – a result which was crucial to Koras-Russell’s proof of the linearization conjecture for Gm -actions on A3. Our method consist in realizing X as an open part of a blowup M −→ A3 and to show that each Ga -action on X descends to A3 . This follows from considerations of the graded algebra associated to O(X ) with respect to a certain filtration. In Paper II, we study Ga-threefolds X which have as their algebraic quotient the affine plane A2 = Sp(C[x, y]) and are a principal bundle above the punctured plane A2 := A2 \ {0}. Equivalently, we study affine Ga -varieties Pˆ that extend a principal bundle P over A2, being P together with an extra fiber over the origin in A2. First the trivial bundle is studied, and some examples of extensions are given (including smooth ones which are not isomorphic to A2 × A). The most basic among the non-trivial principal bundles over A2 is SL2 (C) −→ A2, A 1→ Ae1 where e1 denotes the first unit vector, and we show that any non-trivial bundle can be realized as a pullback of this bundle with respect to a morphism A2 −→ A2. Therefore the attention is then restricted to extensions of SL2(C) and find two families of such extensions via a study of the graded algebras associated with the coordinate rings O(Pˆ) '→ O(P ) with respect to a filtration which is defined in terms of the Ga -actions on P and Pˆ respectively.