We prove an analogue of the fixed-point theorem for the case of definably amenable groups.
We prove an analogue of the fixed-point theorem for the case of definably amenable groups.
Let G be a locally compact group and Γ be any subgroup of G×G,In〔Ⅰ〕Lau and Paterson gave a characterization of inner amenability of G in term of a fixed point …
Let G be a locally compact group and Γ be any subgroup of G×G,In〔Ⅰ〕Lau and Paterson gave a characterization of inner amenability of G in term of a fixed point property for left Banach G module. In this paper we generalize this result to that of Γ-amenability of G.A new fixed point property for amenable grouops is then obtained.
fixed point property. This is the name given by Furstenberg to groups which have a fixed point every time they act affinely on a compact convex set in a locally …
fixed point property. This is the name given by Furstenberg to groups which have a fixed point every time they act affinely on a compact convex set in a locally convex topological linear space. Day [3] has shown that amenability implies the fixed point property. For discrete groups he has shown the converse. Along with amenable groups, we shall study, in this paper, groups with the fixed point property. This paper is based on part of the author's Ph.D. dissertation at Yale University. The author wishes to express his thanks to his adviser, F. J. Hahn. NOTATION. Group will always mean topological group. For a group G, Go will denote the identity component. Likewise Ho will be the identity component of H, etc. Banach spaces, topological vector spaces, etc., will always be over the real field. For topology, we use the notation of Kelley [18], except that our spaces will always
In this paper we describe proofs of the pointwise ergodic theorem and Shannon-McMillan-Breiman theorem for discrete amenable groups, along Følner sequences that obey some restrictions. These restrictions are mild enough …
In this paper we describe proofs of the pointwise ergodic theorem and Shannon-McMillan-Breiman theorem for discrete amenable groups, along Følner sequences that obey some restrictions. These restrictions are mild enough so that such sequences exist for all amenable groups.
We prove the following mean ergodic theorem: for any two commuting measure preserving actions { T g } and { S g } of a countable amenable group G on …
We prove the following mean ergodic theorem: for any two commuting measure preserving actions { T g } and { S g } of a countable amenable group G on a probability space ( X, A , μ), lim n →∞ 1/|Φ n | Σ g ∈Φ n φ( T g x )ψ( S g T g x ) exist in L 1 ( X, A , μ) for any φ, ψ, ∈ L 2 ( X, A , μ), where {Φ n } is any left Følner sequence for G . This generalizes Furstenberg's ergodic Roth theorem, which corresponds to the case G = Z , T g = S g , as well as a more general result of Conze and Lesigne (which corresponds to the case G = Z with no restrictions on T g and S g ). The limit is identified, and two combinatorial corollaries are obtained. The first of these states that in any subset E ⊂ G × G which is of positive upper density (with regard to any left Følner sequence in G × G ), we may find triangular configurations of the form {( a, b ), ( ga, b ), ( ga, gb )}. This result has as corollaries Roth's theorem on arithmetic progressions of length three and a theorem of Brown and Buhler guaranteeing solutions to the equation x + y = 2 z in any sufficiently big subset of an abelian group of odd order. The second corollary states that if G × G × G is partitioned into finitely many cells, one of these cells contains configurations of the form {( a, b, c ), ( ga, b, c ,), ( ga, gb, c ), ( ga, gb, gc )}.
This text is an introduction to de finably amenable NIP groups. It is based on a number of papers, mainly \[6], \[7] and \[4]. This subject has two origins, the …
This text is an introduction to de finably amenable NIP groups. It is based on a number of papers, mainly \[6], \[7] and \[4]. This subject has two origins, the fi rst one is the theory of stable groups and in particular generic types, which were fi rst defi ned by Poizat (see \[12]) and have since played a central role throughout stability theory. Later, part of the theory was generalized to groups in simple theories, where generic types are de ned as types, none of whose translates forks over the empty set.
We study definably amenable NIP groups. We develop a theory of generics, showing that various definitions considered previously coincide, and study invariant measures. Applications include: characterization of regular ergodic measures, …
We study definably amenable NIP groups. We develop a theory of generics, showing that various definitions considered previously coincide, and study invariant measures. Applications include: characterization of regular ergodic measures, a proof of the conjecture of Petrykowski connecting existence of bounded orbits with definable amenability in the NIP case, and the Ellis group conjecture of Newelski and Pillay connecting the model-theoretic connected component of an NIP group with the ideal subgroup of its Ellis enveloping semigroup.
We study definably amenable NIP groups. We develop a theory of generics showing that various definitions considered previously coincide, and we study invariant measures. As applications, we characterize ergodic measures, …
We study definably amenable NIP groups. We develop a theory of generics showing that various definitions considered previously coincide, and we study invariant measures. As applications, we characterize ergodic measures, give a proof of the conjecture of Petrykowski connecting existence of bounded orbits with definable amenability in the NIP case, and prove the Ellis group conjecture of Newelski and Pillay connecting the model-theoretic connected component of an NIP group with the ideal subgroup of its Ellis enveloping semigroup.
We study definably amenable NIP groups. We develop a theory of generics, showing that various definitions considered previously coincide, and study invariant measures. Applications include: characterization of regular ergodic measures, …
We study definably amenable NIP groups. We develop a theory of generics, showing that various definitions considered previously coincide, and study invariant measures. Applications include: characterization of regular ergodic measures, a proof of the conjecture of Petrykowski connecting existence of bounded orbits with definable amenability in the NIP case, and the Ellis group conjecture of Newelski and Pillay connecting the model-theoretic connected component of an NIP group with the ideal subgroup of its Ellis enveloping semigroup.
A series of recent papers by Bergfalk, Lupini and Panagiotopoulus developed the foundations of a field known as `definable algebraic topology,' in which classical cohomological invariants are enriched by viewing …
A series of recent papers by Bergfalk, Lupini and Panagiotopoulus developed the foundations of a field known as `definable algebraic topology,' in which classical cohomological invariants are enriched by viewing them as groups with a Polish cover. This allows one to apply techniques from descriptive set theory to the study of cohomology theories. In this paper, we will establish a `definable' version of a classical theorem from obstruction theory, and use this to study the potential complexity of the homotopy relation on the space of continuous maps $C(X, |K|)$, where $X$ is a locally compact Polish space, and K is a locally finite countable simplicial complex. We will also characterize the Solecki Groups of the Cech cohomology of X, which are the canonical chain of subgroups with a Polish cover that are least among those of a given complexity.
We study definably amenable NIP groups. We develop a theory of generics, showing that various definitions considered previously coincide, and study invariant measures. Applications include: characterization of regular ergodic measures, …
We study definably amenable NIP groups. We develop a theory of generics, showing that various definitions considered previously coincide, and study invariant measures. Applications include: characterization of regular ergodic measures, a proof of the conjecture of Petrykowski connecting existence of bounded orbits with definable amenability in the NIP case, and the Ellis group conjecture of Newelski and Pillay connecting the model-theoretic connected component of an NIP group with the ideal subgroup of its Ellis enveloping semigroup.
We study the action of $G = {\rm SL} (2,\mathbb R)$, viewed as a group definable in the structure $M = (\mathbb R,+,\times )$, on its type space $S_{G}(M)$. We …
We study the action of $G = {\rm SL} (2,\mathbb R)$, viewed as a group definable in the structure $M = (\mathbb R,+,\times )$, on its type space $S_{G}(M)$. We identify a minimal closed $G$-flow $I$ and an idempotent $r\in I$ (with respect to the Ellis se