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1.
We give a combinatorial equivalent to the existence of a non-free hereditarily separable group of cardinality ?1. This can be used, together with a known combinatorial equivalent of the existence of a non-free Whitehead group, to prove that it is consistent that every Whitehead group is free but not every hereditarily separable group is free. We also show that the fact that ? is a p.i.d. with infinitely many primes is essential for this result. 相似文献
2.
Damien Gaboriau 《Journal of Functional Analysis》2011,260(2):414-427
We show that every non-amenable free product of groups admits free ergodic probability measure preserving actions which have relative property (T) in the sense of S. Popa (2006) [Pop06, Def. 4.1]. There are continuum many such actions up to orbit equivalence and von Neumann equivalence, and they may be chosen to be conjugate to any prescribed action when restricted to the free factors. We exhibit also, for every non-amenable free product of groups, free ergodic probability measure preserving actions whose associated equivalence relation has trivial outer automorphisms group. This gives, in particular, the first examples of such actions for the free group on 2 generators. 相似文献
3.
Damien Gaboriau 《Inventiones Mathematicae》2000,139(1):41-98
We study a new dynamical invariant for dicrete groups: the cost. It is a real number in {1−1/n}∪[1,∞], bounded by the number of generators of the group, and it is well behaved with respect to finite index subgroups.
Namely, the quantities 1 minus the cost are related by multiplying by the index. The cost of every infinite amenable group
equals 1. We compute it in some other situations, including free products, free products with amalgamation and HNN-extensions
over amenable groups and for direct product situations. For instance, the cost of the free group on n generators equals n. We prove that each possible finite value of the cost is achieved by a finitely generated group. It is dynamical because
it relies on measure preserving free actions on probability Borel spaces. In most cases, groups have fixed price, which implies that two freely acting groups which define the same orbit partition must have the same cost. It enables us
to distinguish the orbit partitions of probability-preserving free actions of free groups of different ranks. At the end of
the paper, we give a mercuriale, i.e. a list of costs of different groups. The cost is in fact an invariant of ergodic measure-preserving equivalence relations
and is defined using graphings. A treeing is a measurable way to provide every equivalence class (=orbit) with the structure of a simplicial tree, this an example
of graphing. Not every relation admits a treeing: we prove that every free action of a cost 1 non-amenable group is not treeable,
but we prove that subrelations of treeable relations are treeable. We give examples of relations which cannot be produced
by an action of any finitely generated group. The cost of a relation which can be decomposed as a direct product is shown
to be 1. We define the notion for a relation to be a free product or an HNN-extension and compute the cost for the resulting
relation from the costs of the building blocks. The cost is also an invariant of the pairs von Neumann algebra/Cartan subalgebra.
Oblatum 27-I-1999 & 4-IV-1999 / Published online: 22 September 1999 相似文献
4.
Hanfeng Li 《Journal of Functional Analysis》2009,256(10):3368-2350
We show that for any co-amenable compact quantum group A=C(G) there exists a unique compact Hausdorff topology on the set EA(G) of isomorphism classes of ergodic actions of G such that the following holds: for any continuous field of ergodic actions of G over a locally compact Hausdorff space T the map T→EA(G) sending each t in T to the isomorphism class of the fibre at t is continuous if and only if the function counting the multiplicity of γ in each fibre is continuous over T for every equivalence class γ of irreducible unitary representations of G. Generalizations for arbitrary compact quantum groups are also obtained. In the case G is a compact group, the restriction of this topology on the subset of isomorphism classes of ergodic actions of full multiplicity coincides with the topology coming from the work of Landstad and Wassermann. Podle? spheres are shown to be continuous in the natural parameter as ergodic actions of the quantum SU(2) group. We also introduce a notion of regularity for quantum metrics on G, and show how to construct a quantum metric from any ergodic action of G, starting from a regular quantum metric on G. Furthermore, we introduce a quantum Gromov-Hausdorff distance between ergodic actions of G when G is separable and show that it induces the above topology. 相似文献
5.
Abstract We discuss the prospects for finding a “core class,” i.e., a well-behaved class of non-free abelian groups of cardinality ?1 such that every non-free abelian group of cardinality ?1 has a subgroup in the core class. 相似文献
6.
Rostislav Grigorchuk Dmytro Savchuk 《Proceedings of the Steklov Institute of Mathematics》2016,292(1):94-111
We prove a general result about the decomposition into ergodic components of group actions on boundaries of spherically homogeneous rooted trees. Namely, we identify the space of ergodic components with the boundary of the orbit tree associated with the action, and show that the canonical system of ergodic invariant probability measures coincides with the system of uniform measures on the boundaries of minimal invariant subtrees of the tree. Special attention is paid to the case of groups generated by finite automata. Few examples, including the lamplighter group, Sushchansky group, and so-called universal group, are considered in order to demonstrate applications of the theorem. 相似文献
7.
8.
Shigeru Mase 《Annals of the Institute of Statistical Mathematics》1975,27(1):289-298
We shall consider the problem of characterizing infinitely divisible characteristic functions which have only infinitely divisible factors. Infinitely divisible characteristic functions treated in this paper are those which have absolutely continuous Poisson spectral measures and have no Gaussian component in their Lévy canonical representations. It will be shown that Ostrovskii's sufficient condition is also necessary in this case. 相似文献
10.
11.
We answer a question of M. Flach by showing that there is a linear representation of a profinite group whose (unrestricted)
universal deformation ring is not a complete intersection. We show that such examples arise in arithmetic in the following
way. There are infinitely many real quadratic fields F for which there is a mod 2 representation of the Galois group of the maximal unramified extension of F whose universal deformation ring is not a complete intersection. Finally, we discuss bounds on the singularities of universal
deformation rings of representations of finite groups in terms of the nilpotency of the associated defect groups.
The first author was supported in part by NSF Grant DMS01-39737 and NSA Grant H98230-06-1-0021. The second author was supported
in part by NSF Grants DMS00-70433 and DMS05-00106. 相似文献
12.
We prove existence of (at most denumerable many) absolutely continuous invariant probability measures for random one-dimensional dynamical systems with asymptotic expansion. If the rate of expansion (Lyapunov exponents) is bounded away from zero, we obtain finitely many ergodic absolutely continuous invariant probability measures, describing the asymptotics of almost every point. We also prove a similar result for higher-dimensional random non-uniformly expanding dynamical systems. The results are consequences of the construction of such measures for skew-products with essentially arbitrary base dynamics and asymptotic expansion along the fibers. In both cases our method deals with either critical o singular points for the random maps. 相似文献
13.
A. S. Kechris 《Journal of Mathematical Sciences》2007,140(3):398-425
We call a measure-preserving action of a countable discrete group on a standard probability space tempered if the associated
Koopman representation restricted to the orthogonal complement to the constant functions is weakly contained in the regular
representation. Extending a result of Hjorth, we show that every tempered action is antimodular, i.e., in a precise sense
“orthogonal” to any Borel action of a countable group by automorphisms on a countable rooted tree. We also study tempered
actions of countable groups by automorphisms on compact metrizable groups, where it turns out that this notion has several
ergodic theoretic reformulations and fits naturally in a hierarchy of strong ergodicity properties strictly between ergodicity
and strong mixing. Bibliography:s 25 titles.
Dedicated to Professor Anatoly Vershik on the occasion of his 70th birthday
Published in Zapiski Nauchnykh Seminarov POMI, Vol. 326, 2005, pp. 97–144. 相似文献
14.
The concept of (stable) weak containment for measure-preserving actions of a countable group Γ is analogous to the classical notion of (stable) weak containment of unitary representations. If Γ is amenable then the Rokhlin lemma shows that all essentially free actions are weakly equivalent. However if Γ is non-amenable then there can be many different weak and stable weak equivalence classes. Our main result is that the set of stable weak equivalence classes naturally admits the structure of a Choquet simplex. For example, when this simplex has only a countable set of extreme points but when Γ is a nonamenable free group, this simplex is the Poulsen simplex. We also show that when Γ contains a nonabelian free group, this simplex has uncountably many strongly ergodic essentially free extreme points. 相似文献
15.
In this paper, we study the problem of homoclinic orbits to Aubry sets for time-periodic positive definite Lagrangian systems. We show that there are infinitely many homoclinic orbits to some Aubry set under the conditions that the associated Mather set is uniquely ergodic and the first relative homology group of the projection of this Aubry set is nonzero. 相似文献
16.
Robert J. Zimmer 《Israel Journal of Mathematics》1980,35(4):289-300
We consider three problems concerning cocycles of ergodic group actions: behavior of cohomology under the restriction of an
ergodic semi-simple Lie group action to a lattice subgroup; “compactness” of bounded cocyles; and the relation of relative
almost periodicity to relative discrete spectrum for extensions of ergodic actions. 相似文献
17.
In this paper, we show that central order statistics from strictly stationary and ergodic sequences are strongly consistent estimators of population quantiles provided that the quantiles are unique. We generalize this result to strictly stationary but not necessarily ergodic sequences. We also describe three types of possible asymptotic behavior of central order statistics in the case when the corresponding population quantile is not unique. We give applications of the presented results to linear processes with both absolutely continuous and discrete innovations. 相似文献
18.
R. C. Fabec 《Israel Journal of Mathematics》1981,40(2):175-186
We demonstrate that normal ergodic extensions of group actions are characterized as skew product actions given by cocycles
into locally compact groups. As a consequence, Robert Zimmer’s characterization of normal ergodic group actions is generalized
to the noninvariant case. We also obtain the uniqueness theorem which generalizes the von Neumann Halmos uniqueness theorem
and Zimmer’s uniqueness theorem for normal actions with relative discrete spectrum. 相似文献
19.
Sergey Mozgovoy 《Journal of Pure and Applied Algebra》2009,213(4):430-439
We prove a new formula for the generating function of polynomials counting absolutely stable representations of quivers over finite fields. The case of irreducible representations is studied in more detail. 相似文献
20.
Recently, E.C. Lance extended the pointwise ergodic theorem to actions of the group of integers on von Neumann algebras. Our purpose is to extend other pointwise ergodic theorems to von Neumann algebra context: the Dunford-Schwartz-Zygmund pointwise ergodic theorem, the pointwise ergodic theorem for connected amenable locally compact groups, the Wiener's local ergodic theorem for
+
d
and for general Lie groups. 相似文献