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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.
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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(??,??).
2
Pittau, Lorenzo. "Le produit en couronne libre d'un groupe quantique compact par un groupe quantique d'automorphismes". Thesis, Cergy-Pontoise, 2015. http://www.theses.fr/2015CERG0781/document.
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Dans cette thèse on définit et étudie le produit en couronne libre d'un groupe quantique compact par un groupe quantique d'automorphismes, en généralisant la notion de produit en couronne libre par le groupe quantique symétrique introduit par Bichon.Notre recherche est divisée en deux parties. Dans la première, on définit le produit en couronne libre d'un groupe discret par un groupe quantique d'automorphismes. Ensuite, on montre comment décrire les entrelaceurs de ce nouveau objet à l'aide de partitions non-croisées et décorées; à partir de cela et grâce à un résultat de Lemeux, on déduise les représentations irréductibles et les règles de fusion. Ensuite, on prouve des propriétés des algèbres d'opérateurs associées à ce groupe quantique compact, comme la simplicité de la C*-algèbre réduite et la propriété d'Haagerup de l'algèbre de von Neumann.La deuxième partie est une généralisation de la première. D'abord, on définit la notion de produit en couronne libre d'un groupe quantique compact par un groupe quantique d'automorphismes. Après, on généralise la description des espaces des entrelaceurs donnée dans le cas discret et, en adaptant un résultat d'équivalence monoïdale de Lemeux et Tarrago, on trouve les représentations irréductibles et les règles de fusion. Ensuite, on montre des propriétés de stabilité de l'opération de produit en couronne libre. En particulier, on prouve sous quelles conditions deux produits en couronne libres sont monoïdalment équivalents ou ont le semi-anneau de fusion isomorphe. Enfin, on démontre certaines propriétés algébriques et analytiques du groupe quantique duale et des algèbres d'opérateurs associées à un produit en couronne. Comme dernier résultat, on prouve que le produit en couronne de deux groupes quantiques d'automorphismes est isomorphe à un quotient d'un particulier groupe quantique d'automorphismesIn this thesis, we define and study the free wreath product of a compact quantum group by a quantum automorphism group and, in this way, we generalize the previous notion of free wreath product by the quantum symmetric group introduced by Bichon.Our investigation is divided into two part. In the first, we define the free wreath product of a discrete group by a quantum automorphism group. We show how to describe its intertwiners by making use of decorated noncrossing partitions and from this, thanks to a result of Lemeux, we deduce the irreducible representations and the fusion rules. Then, we prove some properties of the operator algebras associated to this compact quantum group, such as the simplicity of the reduced C*-algebra and the Haagerup property of the von Neumann algebra.The second part is a generalization of the first one. We start by defining the notion of free wreath product of a compact quantum group by a quantum automorphism group. We generalize the description of the spaces of the intertwiners obtained in the discrete case and, by adapting a monoidal equivalence result of Lemeux and Tarrago, we find the irreducible representations and the fusion rules. Then, we prove some stability properties of the free wreath product operation. In particular, we find under which conditions two free wreath products are monoidally equivalent or have isomorphic fusion semirings. We also establish some analytic and algebraic properties of the dual quantum group and of the operator algebras associated to a free wreath product. As a last result, we prove that the free wreath product of two quantum automorphism groups can be seen as the quotient of a suitable quantum automorphism group
3
Bauer, Tilman 1973. "[rho]-compact groups as framed manifolds". Thesis, Massachusetts Institute of Technology, 2002. http://hdl.handle.net/1721.1/8401.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2002.In title on t.p. "[rho]" appears as the lower-case Greek letter.
Includes bibliographical references (p. 57-59).
We describe a natural way to associate to any [rho]-compact group an element of the [rho]-local stable stems, which, applied to the [rho]-completion of a compact Lie group G, coincides with the element represented by the manifold G with its left-invariant framing. To this end, we construct a d-dimensional sphere SG with a stable G-action for every d-dimensional [rho]-compact group G, which generalizes the one-point compactification of the Lie algebra of a Lie group. The homotopy class represented by G is then constructed by means of a transfer map between the Thom spaces of spherical fibrations over BG associated with SG.
by Tilman Bauer.
Ph.D.
4
Elkasapy, Abdelrhman. "Word Maps on Compact Lie Groups". Doctoral thesis, Universitätsbibliothek Leipzig, 2015. http://nbn-resolving.de/urn:nbn:de:bsz:15-qucosa-184066.
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Stolz, Abel. "Linear Approximation of Groups and Ultraproducts of Compact Simple Groups". Doctoral thesis, Universitätsbibliothek Leipzig, 2013. http://nbn-resolving.de/urn:nbn:de:bsz:15-qucosa-125643.
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We derive basic properties of groups which can be approximated with matrices. These include closure of classes of such groups under group theoretic constructions including direct and inverse limits and free products. We show that metric ultraproducts of projective linear groups over fields of different characteristics are not isomorphic. We further prove that the lattice of normal subgroups in ultraproducts of compact simple groups is distributive. It is linearly ordered in the case of finite simple groups or Lie groups of bounded rank.
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Metelichenko, Oleksandr Borisovich. "The isodiametric inequality in locally compact groups". Thesis, University College London (University of London), 2006. http://discovery.ucl.ac.uk/1444835/.
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In this thesis it is shown that the isodiametric inequality fails for Carnot-Caratheodory balls in the Heisenberg group W (n E N). Estimates for the ratio of the volume of a ball to the maximal volume of a set of the same diameter are established in this group, and the set of the maximal volume is also found among all sets of revolution about the vertical axis having the same diameter. Results of the similar nature are obtained in the additive group Rn+1 (n G N) with non-isotropic dilations. Using a connection between the isodiametric problem and the Besicovitch 1/2-problem it is proved that the generalized Besicovitch 1/2-conjecture fails in the Heisenberg group HP (1 < n < 8) of the Hausdorff dimension 2n+2 and the additive group Rn+1 (n £ N) having non-isotropic dilations and integer Hausdorff dimension greater than or equal to n + 2. But the 1-dimensional case is shown to be exceptional - the generalized Besicovitch 1/2-conjecture is true in any locally compact group which is equipped with an invariant metric, its Haar measure and has the Hausdorff dimension 1. A question about the relation among the Hausdorff, the spherical and the centred Hausdorff measures of codimension one restricted to a smooth surface is also investigated in the Heisenberg group M1. It is proved that these measures differ but coincide up to positive constant multiples, estimates for which are found.
7
Trotter, Steven. "Involutive algebras and locally compact quantum groups". Thesis, University of Leeds, 2016. http://etheses.whiterose.ac.uk/16168/.
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In this thesis we will be concerned with some questions regarding involutions on dual and predual spaces of certain algebras arising from locally compact quantum groups. In particular we have the $L^1(\G)$ predual of a von Neumann algebraic quantum group $(L^\infty(\G), \Delta)$. This is a Banach algebra (where the product is given by the pre-adjoint of the coproduct $\Delta$), however in general we cannot make this into a Banach $*$-algebra in such a way that the regular representation is a $*$-homomorphism. We can however find a dense $*$-subalgebra $L^1_\sharp(\G)$ that satisfies this property and is a Banach algebra under a new norm. This was originally considered by Kustermans and Vaes when defining the universal C$^*$-algebraic quantum group, however little else has been studied regarding this algebra in general. In this thesis we study the $L^1_\sharp$-algebra of a locally compact quantum group in this thesis. In particular we show how this has a (not necessarily unique) operator space structure such that this forms a completely contractive Banach algebra, we study some properties for compact quantum groups, we study the object for the compact quantum group $\mathrm{SU}_q(2)$ and we study the operator biprojectivity of the $L^1_\sharp$-algebra. In addition we also briefly study some related properties of $C_0(\G)^*$ and its $*$-subalgebra ${C_0(\G)^*}_\sharp$.
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Gaebler, David. "Toeplitz Operators on Locally Compact Abelian Groups". Scholarship @ Claremont, 2004. https://scholarship.claremont.edu/hmc_theses/163.
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Given a function (more generally, a measure) on a locally compact Abelian group, one can define the Toeplitz operators as certain integral transforms of functions on the dual group, where the kernel is the Fourier transform of the original function or measure. In the case of the unit circle, this corresponds to forming a matrix out of the Fourier coefficients in a particular way. We will study the asymptotic eigenvalue distributions of these Toeplitz operators.
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Yang, Qingde. "Multiresolution analysis on non-abelian locally compact groups". Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1999. http://www.collectionscanada.ca/obj/s4/f2/dsk1/tape9/PQDD_0018/NQ43523.pdf.
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Junod, Fabien. "Unstable Adams operations on ρ-local compact groups". Thesis, University of Aberdeen, 2008. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.531931.
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Let G be any compact connected Lie group and let T ≤ G be a maximal torus. Then for any unstable Adams operation f of degree k, the following diagram commutes up to homotopy «!» And conversely, any map f that makes the above diagram commute must be an unstable Adams operation. Using this characterization, we will construct a self-map of a p-local compact group (S,F,L) in order to define unstable Adams operations on a more general setting. THEOREM. For any p-local compact group (S,F,L) there is a self-equivalence such that the map on the objects when restricted to the identity component of S is a qm-th power map.
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Yasemin, Talu E. "p-groups of automorphisms of compact Riemann surfaces". Thesis, University of Aberdeen, 1993. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.357987.
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McGuire, Alex Bernard. "Partially ordered Abelian groups and linearly compact domains". Thesis, University of Essex, 1992. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.359787.
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Akylzhanov, Rauan. "Lp-Lq Fourier multipliers on locally compact groups". Thesis, Imperial College London, 2018. http://hdl.handle.net/10044/1/60829.
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We study the Lp − Lq boundedness of both spectral and Fourier multi- pliers on general locally compact separable unimodular groups G. As a consequence of the established Fourier multiplier theorem we also derive a spectral multiplier theorem on general locally compact separable uni- modular groups. We then apply it to obtain embedding theorems as well as time-asymptotics for the Lp − Lq norms of the heat kernels for general positive unbounded invariant operators on G. We illustrate the obtained results for sub-Laplacians on compact Lie groups and on the Heisenberg group, as well as for higher order operators. With minor modificaitons, our proofs of Paley-type inequalities and Lp − Lq bounds of Fourier multipliers can be adapted to the setting of compact homogeneous manifolds.
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Mattsson, Tobias. "Abstract Harmonic Analysis on Locally Compact Abelian Groups". Thesis, Uppsala universitet, Analys och sannolikhetsteori, 2018. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-354740.
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Rumbelow, Sam. "Pseudodifferential operators on compact abelian groups with applications". Thesis, Swansea University, 2006. https://cronfa.swan.ac.uk/Record/cronfa42386.
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Pseudodifferential operators on compact groups are discussed, with an emphasis on the conditions for which the theorem of Hille and Yosida holds. Some preliminary functional analysis is given including the notion of regularly dissipative operators and Pontrjagin duality. The dual group is described, especially that it is discrete. Some important inequalities, such as Young's inequality, are also stated. Generalised trigonometrical polynomials and generalised Sobolev spaces are defined on the compact group G. A finite exhaustion of the dual space is used to define pointwise convergence and to give a condition for which a generalised Sobolev space is continuously embedded in C(G) and compactly embedded into a larger Sobolev space. The thesis defines k-ellipticity, k-smoothing operators and the k-parametrix, and proves their relation to the compactness of the embedding. It is shown that k-ellipticity is characterised by an inequality of Garding type. Some examples of pseudodifferential operators with constant coefficients are given. Another inequality of Garding type is proved for pseudodifferential operators with variable coefficients, and the existence of a weak solution to (A(x,D) - lambda)u = f is given under certain conditions on the adjoint A*(x,D). A variational solution of B[ϕ,u] = (ϕ,f) is found, and we prove a Garding type inequality for the sesquilinear form.
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Iverson, Joseph. "Frames Generated by Actions of Locally Compact Groups". Thesis, University of Oregon, 2016. http://hdl.handle.net/1794/20443.
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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.
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Cohen, Michael Patrick. "Descriptive Set Theory and Measure Theory in Locally Compact and Non-locally Compact Groups". Thesis, University of North Texas, 2013. https://digital.library.unt.edu/ark:/67531/metadc271792/.
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In this thesis we study descriptive-set-theoretic and measure-theoretic properties of Polish groups, with a thematic emphasis on the contrast between groups which are locally compact and those which are not. The work is divided into three major sections. In the first, working jointly with Robert Kallman, we resolve a conjecture of Gleason regarding the Polish topologization of abstract groups of homeomorphisms. We show that Gleason's conjecture is false, and its conclusion is only true when the hypotheses are considerably strengthened. Along the way we discover a new automatic continuity result for a class of functions which behave like but are distinct from functions of Baire class 1. In the second section we consider the descriptive complexity of those subsets of the permutation group S? which arise naturally from the classical Levy-Steinitz series rearrangement theorem. We show that for any conditionally convergent series of vectors in Euclidean space, the sets of permutations which make the series diverge, and diverge properly, are ?03-complete. In the last section we study the phenomenon of Haar null sets a la Christensen, and the closely related notion of openly Haar null sets. We identify and correct a minor error in the proof of Mycielski that a countable union of Haar null sets in a Polish group is Haar null. We show the openly Haar null ideal may be distinct from the Haar null ideal, which resolves an uncertainty of Solecki. We show that compact sets are always Haar null in S? and in any countable product of locally compact non-compact groups, which extends the domain of a result of Dougherty. We show that any countable product of locally compact non-compact groups decomposes into the disjoint union of a meager set and a Haar null set, which gives a partial positive answer to a question of Darji. We display a translation property in the homeomorphism group Homeo+[0,1] which is impossible in any non-trivial locally compact group. Other related results are peppered throughout.
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Vergara, Soto Ignacio. "Multipliers and approximation properties of groups". Thesis, Lyon, 2018. http://www.theses.fr/2018LYSEN042/document.
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Cette thèse porte sur des propriétés d'approximation généralisant la moyennabilité pour les groupes localement compacts. Ces propriétés sont définies à partir des multiplicateurs de certaines algèbres associés aux groupes. La première partie est consacrée à l'étude de la propriété p-AP, qui est une extension de la AP de Haagerup et Kraus au cadre des opérateurs sur les espaces Lp. Le résultat principal dit que les groupes de Lie simples de rang supérieur et de centre fini ne satisfont p-AP pour aucun p entre 1 et l'infini. La deuxième partie se concentre sur les multiplicateurs de Schur radiaux sur les graphes. L'étude de ces objets est motivée par les liens avec les actions de groupes discrets et la moyennabilité faible. Les trois résultats principaux donnent des conditions nécessaires et suffisantes pour qu'une fonction sur les nombres naturels définisse un multiplicateur radial sur des différentes classes de graphes généralisant les arbres. Plus précisément, les classes de graphes étudiées sont les produits d'arbres, les produits de graphes hyperboliques et les complexes cubiques CAT(0) de dimension finieThis thesis focusses on some approximation properties which generalise amenability for locally compact groups. These properties are defined by means of multipliers of certain algebras associated to the groups. The first part is devoted to the study of the p-AP, which is an extension of the AP of Haagerup and Kraus to the context of operators on Lp spaces. The main result asserts that simple Lie groups of higher rank and finite centre do not satisfy p-AP for any p between 1 and infinity. The second part concentrates on radial Schur multipliers on graphs. The study of these objects is motivated by some connections with actions of discrete groups and weak amenability. The three main results give necessary and sufficient conditions for a function of the natural numbers to define a radial multiplier on different classes of graphs generalising trees. More precisely, the classes of graphs considered here are products of trees, products hyperbolic graphs and finite dimensional CAT(0) cube complexes
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Ghatei, Davoud. "Spin covers of maximal compact subgroups of Kac-Moody groups and of Weyl groups". Thesis, University of Birmingham, 2015. http://etheses.bham.ac.uk//id/eprint/6027/.
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The maximal compact subgroup SO(\(n\),ℝ) of SL(\(n\),ℝ) admits two double covers Spin (\(n\),ℝ) and O(\(n\),ℝ). In this thesis we show that in fact given any simply-laced diagram ∆, with associated split real Kac-Moody group \(G\)(∆), the 'maximal compact' subgroup \(K\)(∆)also admits a double cover Spin(∆). We then describe a subgroup of Spin(∆) which is a cover of the Weyl group \(W\)(∆) associated to g(∆). This answers a conjecture of Damour and Hillmann. Moreover, the group \(D\) which extends \(W\) is used in an attempt to classify the images of the so-called generalised spin representations.
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Watson, Paul Daniel. "Symmetries and automorphisms of compact Riemann surfaces". Thesis, University of Southampton, 1995. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.294489.
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Le, Boudec Adrien. "Géométrie des groupes localement compacts. Arbres. Action !" Thesis, Paris 11, 2015. http://www.theses.fr/2015PA112036.
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Dans le Chapitre 1 nous étudions les groupes localement compacts lacunaires hyperboliques. Nous caractérisons les groupes ayant un cône asymptotique qui est un arbre réel et dont l'action naturelle est focale. Nous étudions également la structure des groupes lacunaires hyperboliques, et montrons que dans le cas unimodulaire les sous-groupes ne satisfont pas de loi. Nous appliquons au Chapitre 2 les résultats précédents pour résoudre le problème de l'existence de points de coupure dans un cône asymptotique dans le cas des groupes de Lie connexes. Dans le Chapitre 3 nous montrons que le groupe de Neretin est compactement présenté et donnons une borne supérieure sur sa fonction de Dehn. Nous étudions également les propriétés métriques du groupe de Neretin, et prouvons que certains sous-groupes remarquables sont quasi-isométriquement plongés. Nous étudions dans le Chapitre 4 une famille de groupes agissant sur un arbre, et dont l'action locale est prescrite par un groupe de permutations. Nous montrons entre autres que ces groupes ont la propriété (PW), et exhibons des groupes simples au sein de cette famille. Dans le Chapitre 5 nous introduisons l'éventail des relations d'un groupe de type fini, qui est l'ensemble des longueurs des relations non engendrées par des relations plus courtes. Nous établissons un lien entre la simple connexité d'un cône asymptotique et l'éventail des relations du groupe, et donnons une grande classe de groupes dont l'éventail des relations est aussi grand que possibleIn Chapter 1 we investigate the class of locally compact lacunary hyperbolic groups. We characterize locally compact groups having one asymptotic cone that is a real tree and whose natural isometric action is focal. We also study the structure of lacunary hyperbolic groups, and prove that in the unimodular case subgroups cannot satisfy a law. We apply the previous results in Chapter 2 to solve the problem of the existence of cut-points in asymptotic cones for connected Lie groups. In Chapter 3 we prove that Neretin's group is compactly presented and give an upper bound on its Dehn function. We also study metric properties of Neretin's group, and prove that some remarkable subgroups are quasi-isometrically embedded. In Chapter 4 we study a family of groups acting on a tree, and whose local action is prescribed by some permutation group. We prove among other things that these groups have property (PW), and exhibit some simple groups in this family. In Chapter 5 we introduce the relation range of a finitely generated group, which is the set of lengths of relations that are not generated by relations of smaller length. We establish a link between simple connectedness of asymptotic cones and the relation range of the group, and give a large class of groups having a relation range as large as possible
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Lozano, Bagén Toni. "On the exoticness of some new p-local compact groups". Doctoral thesis, Universitat Autònoma de Barcelona, 2016. http://hdl.handle.net/10803/385105.
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En 2003, Broto-Levi-Oliver van introduir el concepte de grup p-local finit, el qual és una generalització dels espais classificadors de grups finits p-completats. Més tard, els mateixos autors van introduir també el concepte de grup p-local compacte, el qual és una generalització dels espais classificadors de grups de Lie p-completats i dels grups p-compactes. Mentre que el concepte de grup p-local finit exòtic està clarament definit, en el cas compacte hi ha vàries famílies de grups que donen lloc a grups p-locals compactes, desdibuixant així la noció d'exoticitat.
En aquesta tesi construim nous exemples de grups p-locals finits exòtics per a cada p >= 5. A més, demostrem que aquests nous exemples són simples en el sentit de que no contenen cap subsistema normal propi no trivial.
Llavors, desenvolupem la teoria dels límits de sistemes de fusió. Demostrem que, per a tota família de sistemes de fusió satisfent certes propietats de compatibiliat, podem construir un sistema de fusió relacionat sobre un grup p-toral discret. A més, provem que aquesta construcció del límit coincideix amb el límit directe des d'un punt de vista categòric sota hipòtesis de saturació.
Utilitzant els nous exemples de grups p-locals finits per a p >= 5, així com també altres famílies descobertes per Broto-Levi-Oliver i Díaz-Ruiz-Viruel, apliquem la construcció del límit per a obtenir dos nous exemples de sistemes de fusió sobre grups p-torals discrets per a cada p >= 5 i un nou exemple per a p = 3.
Un cop tenim els nous sistemes de fusió, generalitzem un criteri de saturació conegut per a grups p-locals finits al cas compacte. Llavors, utilitzem aquest criteri per demostrar la saturació dels nous exemples que hem creat, donant lloc així a nous exemples de grups p-locals compactes.
Finalment, demostrem que tant el nou exemple de grup 3-local compacte com els dous nous exemples de grups p-locals compactes per a p >= 5 no es poden realitzar amb grups de Lie compactes ni amb grups p-compactes.In 2003, Broto-Levi-Oliver introduced the concept of p-local finite group, which is a generalization for p-completed classifying spaces of finite groups. Later, the same authors introduced also the notion of p-local compact group, which is a generalization for p-completed classifying spaces of compact Lie groups and p-compact groups. While the concept of exotic p-local finite group is clearly defined, in the compact case there are several families of groups which give rise to p-local compact groups, blurring this way the notion of exoticness. In this thesis we construct new examples of exotic p-local finite groups for every p >= 5. Moreover, we prove that these new examples are simple in the sense that they contain no proper nontrivial normal subsystems. Then, we develop the theory of limits of fusion systems. We prove that, for any family of fusion systems satisfying certain compatibility properties, we can construct a related fusion system over a discrete p-toral group. Moreover, we prove that this limit construction coincides with the direct limit from a categorical point of view under saturation hypothesis. Using the new examples of p-local finite groups for p >= 5, as well as other families of examples discovered by Broto-Levi-Oliver and Díaz-Ruiz-Viruel, we apply the limit construction to produce two new examples of fusion systems over discrete p-toral groups for each p >= 5 and one new example for p = 3. Once we have the new saturated fusion systems, we generalize a saturation criterion known for p-local finite groups to the compact case. Then, we use this criterion to prove the saturation of the new examples we have created, giving rise in this way to new examples of p-local compact groups. Finally, we prove that neither the new example of 3-local compact group nor the new two examples of p-local compact groups for p >= 5 can be realized by compact Lie groups or by p-compact groups.
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Solomyak, Margarita. "Essential spanning forests and electric networks in groups /". Thesis, Connect to this title online; UW restricted, 1997. http://hdl.handle.net/1773/5767.
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Bär, Christian. "Elliptische Operatoren und Darstellungstheorie kompakter Gruppen". Bonn : [s.n.], 1993. http://catalog.hathitrust.org/api/volumes/oclc/31453453.html.
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Krusch, Elvira. "Investigation of the dwarf galaxy population in Hickson compact groups". [S.l.] : [s.n.], 2003. http://deposit.ddb.de/cgi-bin/dokserv?idn=970841698.
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Johnstone, Stephen. "Theory and applications of Fourier multipliers on locally compact groups". Thesis, University of Strathclyde, 2007. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.443141.
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Evens, Samuel R. (Samuel Robert). "Transfer for compact lie groups, induced representations, and braid relations". Thesis, Massachusetts Institute of Technology, 1988. http://hdl.handle.net/1721.1/80453.
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Gül, Erdal. "Controllability of linear systems on non-abelian compact lie groups". Pontificia Universidad Católica del Perú, 2014. http://repositorio.pucp.edu.pe/index/handle/123456789/96429.
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In this paper, we shall deal with a linear control system ∑ defined on a Lie group G with Lie algebra L(G). We prove that, if G is a compact connected Lie group, then the vector fields associated to dynamic of ∑ are conservative, and that if G is also non-Abelian then, by using Poincare Theorem, ∑ is transitive if and only if it is controllable.
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Cave, Christopher. "On coarse geometric properties of discrete and locally compact groups". Thesis, University of Southampton, 2015. https://eprints.soton.ac.uk/384182/.
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Biller, Harald. "Actions of compact groups on spheres and on generalized quadrangles". [S.l. : s.n.], 1999. http://www.bsz-bw.de/cgi-bin/xvms.cgi?SWB8189670.
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Yu, Jun. "Symmetric subgroups of automorphism groups of compact simple Lie algebras /". View abstract or full-text, 2009. http://library.ust.hk/cgi/db/thesis.pl?MATH%202009%20YU.
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Gromada, Daniel [Verfasser]. "Compact matrix quantum groups and their representation categories / Daniel Gromada". Saarbrücken : Saarländische Universitäts- und Landesbibliothek, 2020. http://d-nb.info/1219068810/34.
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Mamino, Marcello. "Topological results on definably compact groups in o-minimal structures". Doctoral thesis, Scuola Normale Superiore, 2010. http://hdl.handle.net/11384/85665.
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Walter, Boris [Verfasser]. "Weighted diffeomorphism groups of Banach spaces and non-compact manifolds and weighted mapping groups / Boris Walter". Paderborn : Universitätsbibliothek, 2014. http://d-nb.info/1058165399/34.
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Crespo, Jonathan. "Monoidal equivalence of locally compact quantum groups and application to bivariant K-theory". Thesis, Clermont-Ferrand 2, 2015. http://www.theses.fr/2015CLF22621/document.
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Les travaux présentés dans cette thèse concernent l'équivalence monoïdale de groupes quantiques localement compacts et ses applications. Nous généralisons au cas localement compact et régulier, deux résultats importants concernant les actions de groupes quantiques compacts. Soient G1 et G2 deux groupes quantiques localement compacts réguliers et monoïdalement équivalents. Nous développons un procédé d'induction des actions qui permet d'établir une équivalence canonique des catégories dont les objets sont les actions continues de G1 et G2 sur les C*-algèbres. Comme application de ce résultat, nous obtenons une équivalence canonique des catégories de KK-Théorie équivariante pour G1 et G2. Nous introduisons et étudions une notion d'actions sur les C*-algèbres, de groupoïdes quantiques mesurés sur une base finie. La preuve de la seconde équivalence s'appuie alors sur une version du théorème de bidualité de Takesaki-Takai pour les actions de groupoïdes quantiques mesurés sur une base finie. Enfin, nous terminons en définissant et étudiant une notion de modules hilbertiens équivariants pour des actions de groupoïdes quantiques mesurés sur une base finieThis dissertation deals with the notion of monoidal equivalence of locally compact quantum groups and its applications. We generalize to the case of regular locally compact quantum groups, two important resultst concerning the actions of compact quantum groups. Let G1 and G2 be two regular locally compact quantum groups monoidally equivalent. We develop an induction procedure and we build an equivalence of the categories, whose objects are the continuous actions of G1 and G2 on C*-algebras. As an application of this result, we obtain a canonical equivalence of the categories of equivariant KK-theory for actions of G1 and G2. We introduce and investigate a notion of actions on C*-algebras of mesured quantum groupoids on a finite basis. The proof of the second equivalence relies on a version of the Takesaki-Takai duality theorem for continuous actions of measured quantum groupoids on a finite basis. We conclude by defining and studying a notion of equivariant Hilbert modules for actions of mesured quantum groupoids on a finite basis
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Renedo, Marco Antonio. "Mixing properties of actions by commuting epimorphisms of Compact Abelian Groups /". The Ohio State University, 1998. http://rave.ohiolink.edu/etdc/view?acc_num=osu1487953567771096.
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Kazeem, Funmilayo Eniola. "A measure for the number of commuting subgroups in compact groups". Doctoral thesis, Faculty of Science, 2019. http://hdl.handle.net/11427/30383.
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The present thesis is devoted to the construction of a probability measure which counts the pairs of closed commuting subgroups in infinite groups. This measure turns out to be an extension of what was known in the finite case as subgroup commutativity degree and opens a new approach of study for the class of near abelian groups, recently introduced in [24, 27]. The extremal case of probability one characterises the topologically quasihamiltonian groups, studied originally by K. Iwasawa [30, 31] in the abstract case and then by F. K¨ummich [35, 36, 37], C. Scheiderer [45, 46], P. Diaconis [11] and S. Strunkov [48] in the topological case. Our probability measure turns out to be a useful tool in describing the distance of a profinite group from being topologically quasihamiltonian. We have been inspired by an idea of H. Heyer in the present context of investigation and in fact we generalise some of his techniques, in order to construct a probability measure on the space of closed subgroups of a profinite group. This has been possible because the space of closed subgroups of a profinite group may be approximated by finite spaces and the consequence is that our probability measure may be approximated by finite probability measures. While we have a satisfactory description for profinite groups and compact groups, the case of locally compact groups remains open in its generality.
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Maher, David Graham School of Mathematics UNSW. "Brownian motion and heat kernels on compact lie groups and symmetric spaces". Awarded by:University of New South Wales. School of Mathematics, 2006. http://handle.unsw.edu.au/1959.4/28295.
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An important object of study in harmonic analysis is the heat equation. On a Euclidean space, the fundamental solution of the associated semigroup is known as the heat kernel, which is also the law of Brownian motion. Similar statements also hold in the case of a Lie group. By using the wrapping map of Dooley and ildberger, we show how to wrap a Brownian motion to a compact Lie group from its Lie algebra (viewed as a Euclidean space) and find the heat kernel. This is achieved by considering It??o type stochastic differential equations and applying the Feynman-Ka??c theorem. We also consider wrapping Brownian motion to various symmetric spaces, where a global generalisation of Rouvi`ere???s formula and the e-function are considered. Additionally, we extend some of our results to complex Lie groups, and certain non-compact symmetric spaces.
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O'Brien, Ellen Marie. "Finitely asymptotic properties of powers of characters of compact semisimple Lie groups". Thesis, University of Ottawa (Canada), 1994. http://hdl.handle.net/10393/10307.
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This thesis has to do with decomposing tensor powers of irreducible representations of compact semisimple Lie groups or their Lie algebras. We will be concerned only with the set of irreducible representations appearing in the decomposition. In particular, we determine whether, for a sufficiently high tensor power, this set is "maximal", in a sense to be made more precise below. We rely on the theory of semisimple Lie algebras, and describe the results in terms of characters. A (reducible) character of a finite dimensional complex semisimple Lie algebra is saturated if for each of its dominant weights, the corresponding irreducible character is a summand in its orthogonal decomposition. A character is eventually saturated if some power is saturated. In the first part of the thesis, we describe all eventually saturated irreducible characters. In particular, it is shown that any irreducible character whose highest weight is in the interior of the Weyl chamber is eventually saturated; if the Lie algebra is simply laced, then all irreducible characters are eventually saturated; if not, then there are irreducible characters (with highest weights on the boundary of the Weyl chamber) that are not eventually saturated. We use the PRV conjecture to derive necessary and sufficient conditions for eventual saturation. These conditions are expressed in terms of cones generated by the weights of the character. The convex hull of the weight diagram, and the weights adjacent to the highest weight along the edges of the convex hull, are described in detail. We show in the second part of the thesis that if the Lie algebra is A$\sb{d}$, and $d \ \le$ 5, then for any integer $n \ge \ d$ + 1 and any irreducible character $\chi\lambda$, the product $\chi\sbsp{\lambda}{n}$ is saturated. We also establish this result for certain irreducible characters of $A\sb{d}, d \ge$ 5. These results are proved by induction on the rank of the Lie algebra and on the highest weight of the character. The geometry of the convex hull of the set of weights comes in to play here as well, and the dominant faces of this set are described. A reduction result, relating the "restriction to a dominant face" of the decomposition of a product of characters to that of a corresponding product in an algebra of lower rank, is established. The Littlewood-Richardson rule is used to compare products in which the component irreducible characters have highest weights which differ by a small amount. Similar induction arguments are used to describe all the irreducible characters appearing in the decomposition of a product of irreducible characters of A$\sb2$. Some of the irreducible characters appearing in a product of characters of higher rank algebras can also be determined using this type of induction. The questions considered here arise in the study of product type actions of compact groups. The results on eventual saturation of irreducible characters are useful in computing the equivariant ordered K-theory of certain fixed point C$\sp*$-algebras under the corresponding group actions.
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Vassar, Susan. "A study of compact groups selected from the APM Bright Galaxy Catalogue". Thesis, University of Oxford, 1994. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.260177.
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馮淑貞 i Suk-ching Fung. "Asymptotic vanishing theorem of cohomology groups on compact quotientsof the unit ball". Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 1998. http://hub.hku.hk/bib/B31220848.
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Deutsch, Annabel. "Kazhdan's property (T) and related properties of locally compact and discrete groups". Thesis, University of Edinburgh, 1992. http://hdl.handle.net/1842/13631.
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In this thesis we look at a number of properties related to Kazhdan's property (T), for a locally compact, metrisable, σ-compact group. For such a group, G, the following properties are equivalent. 1. Kazhdan's definition of property (T): the trivial representations is isolated in the unitary dual of G (with the Fell topology). 2. The group, G, is compactly generated and for every compact generating set, K, there is a positive constant, ε, such that if π is a unitary representation of G on a Hilbert space, cal H, and ζ is a unit vector in cal H such that vskip 0.7cmthen π fixes some non-zero vector in cal H. This is often taken as the definition of property (T). 3. Every conditionally negative type function on G is bounded. 4. For a discrete group, G is finitely generated and for every finite generating set, K, zero is an isolated point in the spectrum of the Laplacian. From 2 we can define the Kazhdan constant, the largest possible value of ε for a given G and K. In Chapter 2 we investigate how to calculate these constants. In Chapter 4 we look at the bound on conditionally negative type functions and use its existence to extend a result of A.Connes and V.Jones about the von Neumann algebras of property (T) groups. The first half of Chapter 3 examines the spectrum of the Laplacian for a discrete group and finite generating set and compares its least positive element to the Kazhdan constant. Non-compact property (T) groups are all non-amenable. However, the standard example of a non-amenable group F_2, does not have property (T). The second half of Chapter 3 looks at the spectrum of λ(Δ) for F_2 with various generating sets, where λ is the left regular representation of G on l2(G). For any non-amenable group, the smallest element of Spλ(Δ) is positive. Chapter 5 is an attempt to extend various results about F2 to other non-amenable groups.
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Miles, Richard Craig. "Arithmetic dynamical systems". Thesis, University of East Anglia, 2000. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.323222.
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Wang, Xumin. "Functional and harmonic analysis of noncommutative Lp spaces associated to compact quantum groups". Thesis, Bourgogne Franche-Comté, 2019. http://www.theses.fr/2019UBFCD040.
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Cette thèse a pour but d'étudier l'analyse sur les groupes quantiques compacts. Elle se compose de deux parties. La première présente la classification des semi-groupes de Markov invariants sur ces espaces homogènes quantiques. Les générateurs de ces semi-groupes sont considérés comme des opérateurs de Laplace sur ces espaces.La sphère classique , la sphère libre et la sphère semi-libérée sont considérées comme des exemples et les générateurs de semi-groupes de Markov sur ces sphères sont classés. Nous calculons aussi les dimensions spectrales des trois familles de sphères en fonction du comportement asymptotique des valeurs propres de leur opérateur de Laplace.Dans la deuxième partie, nous étudions la convergence des séries de Fourier pour les groupes non abéliens et les groupes quantiques. Il est bien connu qu'un certain nombre de propriétés d'approximation de groupes peuvent être interprétées comme des méthodes de sommation et de convergence moyenne de séries de Fourier non commutatives associées. Nous établissons un critère général d'inégalités maximales pour les identités approximatives de multiplicateurs non commutatifs de Fourier. En conséquence, nous prouvons que pour tout groupe dénombrable discret moyennable, il existe une suite de fonctions définies positives à support fini, telle que les multiplicateurs de Fourier associés sur les espaces Lp non commutatifs satisfassent à la convergence ponctuelle. Nos résultats s'appliquent également à la convergence presque partout des séries de Fourier de fonctions Lp sur des groupes compacts non-abéliens. D'autre part, nous obtenons des bornes indépendantes de la dimension pour les inégalités maximales de Hardy-Littlewood non commutatives dans l'espace à valeurs opérateurs associées à des corps convexesThis thesis is devoted to studying the analysis on compact quantum groups. It consists of two parts. First part presents the classification of invariant quantum Markov semigroups on these quantum homogeneous spaces. The generators of these semigroups are viewed as Laplace operators on these spaces.The classical sphere, the free sphere, and the half-liberated sphere are considered as examples and the generators of Markov semigroups on these spheres are classified. We compute spectral dimensions for the three families of spheres based on the asymptotic behavior of the eigenvalues of their Laplace operator.In the second part, we study of convergence of Fourier series for non-abelian groups and quantum groups. It is well-known that a number of approximation properties of groups can be interpreted as some summation methods and mean convergence of associated noncommutative Fourier series. We establish a general criterion of maximal inequalities for approximative identities of noncommutative Fourier multipliers. As a result, we prove that for any countable discrete amenable group, there exists a sequence of finitely supported positive definite functions, so that the associated Fourier multipliers on noncommutative Lp-spaces satisfy the pointwise convergence. Our results also apply to the almost everywhere convergence of Fourier series of Lp-functions on non-abelian compact groups. On the other hand, we obtain the dimension free bounds of noncommutative Hardy-Littlewood maximal inequalities in the operator-valued Lp space associated with convex bodies
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Kindl, Enrico. "A photometric and morphological study of compact groups of galaxies and their environments". Thesis, University of British Columbia, 1987. http://hdl.handle.net/2429/30722.
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This thesis examines properties of galaxies in and around compact groups. Astrometry, photometry and morphological classifications are derived from CCD images for all 463 galaxies in Hickson's sample of 100 compact groups. Some minor revisions to the membership of the original catalog are made. At high galactic latitude (b > 30°), the catalog is estimated to be 90% complete for groups with total B[formula omitted] magnitude 13.0 or less. 49% of all the catalogued galaxies, and 48% of first-ranked galaxies are spiral. No significant difference is found between the distribution of morphological types of first-ranked galaxies and all group galaxies. Morphological concordance occurs among galaxies within a group: more galaxies are the same type (spiral or nonspiral), than would be expected by chance. Galaxy morphological type correlates with group optical luminosity and, more strongly, with velocity dispersion, but not with galaxy space density. These results imply that the morphological types of galaxies in compact groups are strongly influenced by the environment, and that this influence occurs mostly at the time of galaxy formation.
Fields surrounding 97 compact groups with known redshifts were examined on the Palomar Observatory Sky Survey prints. 3889 galaxies were identified within 1.125h⁻¹ Mpc of the centre of each group. Coordinates, magnitudes, diameters, and Hubble types are derived for these galaxies. 78% of the groups show no significant excess of field galaxies within 0.5h⁻¹ Mpc. This indicates that most compact groups are truly isolated. 59% of these field galaxies were classified as spiral, a higher fraction that for the group galaxies. This difference is more pronounced for groups which do show a significant field galaxy excess. These results indicate that most
groups are dense dynamical entities.
Monte-Carlo calculations indicate that 35% of galaxy quintets are predicted to contain a single discordant redshift due to the chance alignment of an unrelated field galaxy. This is in agreement with the observed number of four discordant quintets in 10. These results are consistent with the cosmological interpretation of galaxy redshifts.Science, Faculty of
Physics and Astronomy, Department of
Graduate
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Kliemann, Malte [Verfasser]. "The Thickness of Left-Invariant Metrics on Compact Connected Lie Groups / Malte Kliemann". Kiel : Universitätsbibliothek Kiel, 2019. http://d-nb.info/1177797887/34.
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Schneider, Jakob [Verfasser], Andreas [Gutachter] Thom i Martin [Gutachter] Liebeck. "On ultraproducts of compact quasisimple groups / Jakob Schneider ; Gutachter: Andreas Thom, Martin Liebeck". Dresden : Technische Universität Dresden, 2021. http://d-nb.info/1232413577/34.
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Fung, Suk-ching. "Asymptotic vanishing theorem of cohomology groups on compact quotients of the unit ball /". Hong Kong : University of Hong Kong, 1998. http://sunzi.lib.hku.hk/hkuto/record.jsp?B20667991.
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Bywaters, Timothy Peter. "Connections Between Willis' Theory for Totally Disconnected Locally Compact Groups and Graph Automorphisms". Thesis, The University of Sydney, 2019. http://hdl.handle.net/2123/21148.
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We investigate the tidy subgroups, scale function and related invariants of totally disconnected locally compact groups. Our focus is on relating these ideas to combinatorial and geometric aspects of the group. After giving necessary background, we study the scale function and tidy subgroups of an endomorphism of a totally disconnected locally compact group. Our results are inspired by a similar investigation for automorphisms by Möller (Can. J. Math., 54(4), 795-827). We characterise when a compact open subgroup is tidy for an endomorphism in terms of a graph constructed from the subgroups and the endomorphism. Using this characterisation, we develop a tidying procedure which produces from a compact open subgroup, a tidy subgroup. We also use our characterisation to prove a tree representation theorem for endomorphisms, inspired by a similar theorem of Baumgartner and Willis (Isr. J. Math., 142(1), 221-248) for automorphisms. We then study restricted Burger-Mozes groups. These are algebraic subgroups of the autmorphism group of a regular tree but are not equipped with the permutation topology. The stabiliser of a vertex in these groups is open but not compact. We calculate invariants for these groups and relate them to similar calculations done for the automorphism group of a regular tree. This gives insight on how results for the automorphism group of a regular tree may generalise to a larger class of totally disconnected locally compact groups. We investigate the space of directions for a totally disconnected locally compact group acting vertex transitively with compact open vertex stabilisers on a hyperbolic graph. Generalising the discrete case, we call such groups hyperbolic. We show that the space of directions for a hyperbolic group is a discrete metric space and that asymptotic classes are determined by fixed points on the boundary of the hyperbolic graph. This verifies a conjecture of Baumgartner, Möller and Willis (Isr. J. Math., 190(1), 365-388).
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Stolz, Abel [Verfasser], Andreas [Akademischer Betreuer] Thom, Andreas [Gutachter] Thom i Nikolay [Gutachter] Nikolov. "Linear Approximation of Groups and Ultraproducts of Compact Simple Groups / Abel Stolz ; Gutachter: Andreas Thom, Nikolay Nikolov ; Betreuer: Andreas Thom". Leipzig : Universitätsbibliothek Leipzig, 2013. http://d-nb.info/1238527353/34.
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