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Countable Borel Equivalence Relations
 J. MATH. LOGIC
"... This paper is a contribution to a new direction in descriptive set theory that is being extensively pursued over the last decade or so. It deals with the development of the theory of definable actions of Polish groups, the structure and classification of their orbit spaces, and the closely related s ..."
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Cited by 87 (11 self)
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This paper is a contribution to a new direction in descriptive set theory that is being extensively pursued over the last decade or so. It deals with the development of the theory of definable actions of Polish groups, the structure and classification of their orbit spaces, and the closely related study of definable equivalence relations. This study is motivated by basic foundational questions, like understanding the nature of complete classification of mathematical objects, up to some notion of equivalence, by invariants, and creating a mathematical framework for measuring the complexity of such classification problems. (For an extensive discussion of these matters, see, e.g., Hjorth [00], Kechris [99, 00a].) This theory is developed within the context of descriptive set theory, which provides the basic underlying concepts and methods. On the other hand, in view of its broad scope, there are natural interactions of it with other areas of mathematics, such as model theory, recursion theory, the theory of topological groups and their representations, topological dynamics, ergodic theory, and operator algebras
Random Walks On The Affine Group Of Local Fields And Of Homogeneous Trees
, 1993
"... Introduction The starting point for the work presented here is the study of probabilistic and potential theoretic properties of products of random affine transformations, that is, random matrices of the the form \Gamma a b 0 1 \Delta , where a 6= 0. Products of real affine transformations were ..."
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Cited by 28 (13 self)
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Introduction The starting point for the work presented here is the study of probabilistic and potential theoretic properties of products of random affine transformations, that is, random matrices of the the form \Gamma a b 0 1 \Delta , where a 6= 0. Products of real affine transformations were, probably, one of the first examples of random walks on groups where noncommutativity critically influences asymptotic properties of the random walk and leads to essentially new phenomena, see Grenander [Gr]. Later on, the affine group over R always remained one of the first examples to be considered when addressing new problems connected with noncommutative random walks (see Molchanov [Mo], Grincevicjus [G1], [G2], Elie [E1], [E2], [E3] and others), the other typical example being that of free groups (see the survey by Woess [W2] for references). Due to the structure theory of Lie groups
STRICT INEQUALITIES FOR CONNECTIVE CONSTANTS OF TRANSITIVE GRAPHS
"... Abstract. The connective constant of a graph is the exponential growth rate of the number of selfavoiding walks starting at a given vertex. Strict inequalities are proved for connective constants of vertextransitive graphs. Firstly, the connective constant decreases strictly when the graph is repl ..."
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Cited by 1 (0 self)
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Abstract. The connective constant of a graph is the exponential growth rate of the number of selfavoiding walks starting at a given vertex. Strict inequalities are proved for connective constants of vertextransitive graphs. Firstly, the connective constant decreases strictly when the graph is replaced by a nontrivial quotient graph. Secondly, the connective constant increases strictly when a quasitransitive family of new edges is added. These results have the following implications for Cayley graphs. The connective constant of a Cayley graph decreases strictly when a new relator is added to the group, and increases strictly when a nontrivial group element is declared to be a generator. 1.
Construction Of Discrete, NonUnimodular Hypergroups
"... We explain how one can construct a class of discrete hypergroups which are nonunimodular. They arise as double coset hypergroups induced by the transitive action of a nonunimodular group of permutations on an infinite set. A concrete example is given in terms of the affine group of a homogeneous t ..."
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Cited by 1 (1 self)
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We explain how one can construct a class of discrete hypergroups which are nonunimodular. They arise as double coset hypergroups induced by the transitive action of a nonunimodular group of permutations on an infinite set. A concrete example is given in terms of the affine group of a homogeneous tree.
Computations of spectral radii on Gspaces
 CONTEMPORARY MATHEMATICS
"... In previous work, we have developped methods to compute norms and spectral radii of transition operators on proper metric spaces. The operators are assumed to be invariant under a locally compact, amenable group which acts with compact quotient. Here, we present several further applications of thos ..."
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In previous work, we have developped methods to compute norms and spectral radii of transition operators on proper metric spaces. The operators are assumed to be invariant under a locally compact, amenable group which acts with compact quotient. Here, we present several further applications of those methods. The first concerns a generalization of an identity of Hardy, Littlewood and Pólya. The second is a detailed study of a class of diffusion operators on a homogeneous tree, seen as a 1complex. Finally, we investigate the implications of our method for computing spectral radii of convolution operators on general locally compact groups and Lie groups.