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Amenable group - Wikipedia, the free encyclopedia In mathematics, an amenable group is a topological group G carrying a kind of averaging operation, that is invariant under translations by group elements. The definition of amenability is quite a lot simpler in the case of a discrete group, i.e. By the fundamental theorem of finitely generated abelian groups, it follows that abelian groups are amenable. en.wikipedia.org /wiki/Amenable_group   (649 words)

PlanetMath: amenable group All finite groups and all abelian groups are amenable. Compact groups are amenable as the Haar measure is an (unique) invariant mean. This is version 6 of amenable group, born on 2002-11-15, modified 2005-02-02. planetmath.org /encyclopedia/AmenableGroup.html   (138 words)

Thompson groups - Wikipedia, the free encyclopedia In mathematics, the Thompson groups (also called Thompson's groups, vagabond groups or chameleon groups) are three groups, commonly denoted F, T and V, which were first studied by the logician Richard Thompson in 1965. The group F is "just non-abelian" in the sense that it is not abelian, but all its proper homomorphic images are abelian. If it turns out not to be amenable, then it will provide another counterexample to the long-standing but recently disproved von Neumann conjecture for finitely presented groups, which suggested that a finitely presented group is amenable if and only if it does not contain a copy of the free group of rank 2. en.wikipedia.org /wiki/Thompson_groups   (340 words)

[No title] The isometry group of a compact Lorentz manifold, I, with Garrett Stuck, {\it Inventiones Mathematic\ae}, {\bf129} (1997), 239-261. For example, one group may be $\splin_2(\R)$ and the other a group with infinite discrete center ({\it e.g.}, the universal cover of $\splin_2(\R)$); I believe this is the first rigidity result of this type for a pair of simple Lie groups both of split rank one. We also demonstrate that an action is amenable iff it is the Mackey range of a homomorphism from an amenable countable equivalence relation iff it is the Posson boundary of a group-invariant matrix-valued Markov random walk on $G$. www.math.umn.edu /~adams/publ.txt   (2431 words)

[No title]   (Site not responding. Last check: 2007-10-13) But the torsion subgroup in Grigorchuk's group does not have a bounded exponent and his group is amenable (it was the first example of a finitely presented amenable but not elementary amenable group). This immediately implies that ${\mathcal G}$ is an extension of a group of exponent $n$ (the union of the increasing sequence of subgroups ${\bf t}^s\la {\mathcal A}\ra {\bf t}^{-s}, s=1,2,\dots)$ by a cyclic group. This group can be regarded as a factor-group of the group $H'$ generated by ${\mathcal C}$ subject to the relations (1) from ${\mathcal R}$ and derived relations (5) (this group is denoted in the paper by $H_{kra}$) over the normal subgroup generated by Burnside relations (4). maths.tcd.ie /EMIS/journals/ERA-AMS/2001-01-009/2001-01-009.tex.html   (3620 words)

Abstract   (Site not responding. Last check: 2007-10-13) The growth in groups is a subject initiated by A. Schwartz, J. Milnor, J. Wolf and H. The groups of intermediate growth constructed by the speaker in 1983 (as the answer to a question of Milnor) opened a new direction in the study of groups acting on trees including groups generated by finite automata. groups that can be constructed from finite and commutative groups by operations of extensions and direct limits, and secondly as the class NF of groups without free subgroup with two generators. www.math.tamu.edu /research/algcom/abstracts_01-02/011026.html   (279 words)

Annals of Mathematics, II. Series, Vol. 151, No. 3, pp. 1119-1150, 2000 For groups in which there is a convenient notion of past, actions with completely positive entropy are known to be mixing of all orders. For amenable group actions, both properties make sense, but in general there is no natural past to use in developing the entropy theory. The possibility that such methods may be of use in resolving the open problems concerning the spectral theory of amenable group actions with completely positive entropy, and the nature of Følner sequences along which pointwise ergodic theorems may hold, is raised and discussed briefly. www.univie.ac.at /EMIS/journals/Annals/151_3/5.html   (467 words)

[No title]   (Site not responding. Last check: 2007-10-13) Thus we have fulfilled the hypotheses of Paterson's Theorem~2~\cite{Paterson}, hence the unitary group of $C(X,\,M(n))$ with the relative weak topology is an amenable topological group. There has been another known weaker amenable group topology on the group $C^\infty(X,G)$, which is the topology of convergence in measure. Specifically, if the group $G$ is locally compact amenable, then the group of measurable maps from the standard Borel space $X$, equipped with a non-atomic probability measure $\mu,$ to $G$ is {\it extremely amenable,} that is, has a fixed point in every compact space on which it acts continuously cf.~Theorem 2.2 in~\cite{Pestov1}. www.ma.utexas.edu /mp_arc/html/papers/04-14   (3467 words)

Publications of Mark Sapir This group is torsion of exponent n>>1 by cyclic. Here is a paper (joint with Olshanskii) where we present easy quasi-isometric embeddings of relatively free groups in finitely based varieties (in particular, the free Burnside groups) and Baumslag-Solitar groups into finitely presented groups with small Dehn functions. In particular, we proved that a finitely generated group G has word problem in NP if and only if G is a subgroup of a finitely presented group H with polynomial Dehn function (moreover this subgroup has bounded length distortion). atlas.math.vanderbilt.edu /~msapir/publications.html   (1129 words)

Papers.html   (Site not responding. Last check: 2007-10-13) Amenable group actions, the Rusiewicz problem, the Hahn-Banach theorem and Lebesgue measurability. The first three papers are devoted to the question of when the algebraic structure of an amenable group determines the collection of invariant means under actions of the group. The question of the existence of an action of a locally finite group on the integers admitting a unique invariant mean is shown to be independent of Z.F.C. and thus not resolvable by the usual axioms for mathematics (even with the Axiom of Choice). www.math.uci.edu /sub2/Foreman/homepage/newpapers.html   (2671 words)

Left ordered amenable and locally indicable groups   (Site not responding. Last check: 2007-10-13) Recall that the class of elementary amenable groups is the smallest class of groups which contains all abelian-by-finite groups, is closed under group extension, and is closed under directed unions. It is well known that every solvable-by-finite group is elementary amenable, and every elementary amenable group is amenable. It is left as an open problem to as whether every left ordered amenable group is locally indicable. www.math.vt.edu /people/linnell/research/Order   (187 words)

Citebase - Amenability via random walks   (Site not responding. Last check: 2007-10-13) Our results separate the class of amenable groups from the closure of subexponentially growing groups under the operations of group extension and direct limits; these classes are separated even within the realm of finitely presented groups. For locally compact topological groups with this property we show that almost all finite subsets of the group generate free subgroups. Since Powers recognised in 1975 that non-abelian free groups are C*-simple, large classes of groups which appear naturally in geometry have been identified, including non-elementary Gromov hyperbolic groups and lattices... www.citebase.org /cgi-bin/citations?id=oai:arXiv.org:math/0305262   (607 words)

Lie Group Seminar For a lattice \Gamma in a Lie group G, we study the distribution of orbits of \Gamma in a homogeneous space G/H. We show that in many cases, every dense \Gamma-orbit is equidistributed with respect to a smooth measure. However, unlike in the case of amenable group actions, this measure is not \Gamma-invariant, and it depends non-trivially on the starting point. For an example of $\Gamma$ in the automorphism group of a $2p$-regular tree whose quotient is a loop, we obtain a lower bound of the asymptotic behavior as well. www.its.caltech.edu /~gorodnik/lieseminar.html   (1452 words)

Budapest University of Technology verifies in the case of a compact, metrizable   group the existence of a measure, that is not identically zero   and is in consideriton of metric is invariant. According to that, if the group  is locally compact, metrizable, non-discrete and of  finite   dimension, then there is existing a Hausdorff-function, that   the left-side Hausdorff-measure induced by this, would be a  left-side Haar-measure. In the remaining   part of this Chapter, we examine the characteristics of that  topological group on which there is existing a  non-identically zero, left-invariant measure. www.math.bme.hu /~arpi/doktori.htm   (700 words)

Elementary Amenable Groups and 4-Manifolds with Euler Characteristic 0   (Site not responding. Last check: 2007-10-13) Elementary Amenable Groups and 4-Manifolds with Euler Characteristic 0 In particular a finitely presentable group which has a nontrivial elementary amenable subgroup whose finite subgroups have bounded order and with no nontrivial finite normal subgroup must have deficiency at most 1, and if it has a presentation of deficiency 1 then the corresponding 2-complex is aspherical. Similarly if the fundamental group of a closed 4-manifold with Euler characteristic 0 is virtually torsion free and elementary amenable then it either has 2 ends ot is virtually an extension of Z by a subgroup of Q, or the manifold is aspherical and the group is virtually poly-Z of Hirsch length 4. anziamj.austms.org.au /JAMSA/V50/Part1/Hillman.html   (160 words)

UIUC Dept. of Mathematics Seminar Calendar Abstract: For any finitely generated group $G$, we define $\fol G$ as the infimum over all the finite sets of generators $X$ of $G$ and over all finite subsets $A\subset G$, of the ratio between the cardinals of the $X$-boundary of $A$, and $A$. We prove that the following classes of groups are uniformly non-amenable: non-abelian free groups, non-elementary word-hyperbolic groups, large groups, free Burnside groups of large enough odd exponent, and groups acting acylindrically on a tree. Finally, we exhibit a family of non-amenable groups (in particular including all non-solvable Baumslag-Solitar groups) which are not uniformly non-amenable, that is, they satisfy $\fol G=0$. torus.math.uiuc.edu /cal/math/cal?year=2004&month=04&day=01&interval=day   (422 words)

Faculty: Masamichi Takesaki   (Site not responding. Last check: 2007-10-13) Takesaki has worked on the structure analysis of the automorphism and the unitary groups of an AFD factor and amenable discrete group actions on it in collaboration with various mathematicians including Y. Katayama, Y. Kawahigashi, S. Popa and C.E. Sutherland. Kawakigashi, C.E. Sutherland and M. Takesaki, The structure of the automorphism group of an injective factors and cocycle conjugacy of discrete abelian group actions, Acta Math 169 (1992), 105-130. Katayama, C.E. Sutherland and M. Takesaki, The characteristic square of a factor and the cocyle conjugacy of disdrete group actins on factors, Inventiones Mathematicae, 132 (1998), 331-380. www.math.ucla.edu /faculty/mt.html   (241 words)

Untitled Document There is a natural surjection from a certain hyperbolic Coxter group onto the mapping class group of a genus 5 surface for which the image of the Coxeter element is Leininger’s example realizing Lehmer’s number, indicating some connection between the eigenvalues of elements of Coxeter groups and pseudo-Anosov dilatation factors. He gave a result of Gromov which characterizes when the group is trivial or else word hyperbolic, according to a fixed “density” value in the chosen “randomness” or density model. Conversely, Shalom also showed that some features of certain groups which are, a priori, geometric in nature, such as having a uniformly embedded amenable group, yield a lower bound on the rational cohomological dimension. www.math.cornell.edu /~festival/2003/report.html   (1482 words)

UIUC Dept. of Mathematics Seminar Calendar Abstract: The Maier-Schmidt Theorem asserts that a core-free permutable subgroup of a finite group is contained in the hypercenter. Roughly speaking, equivariant cohomology allows one to consider the cohomology of a space relative to a group action (for us, a compact Lie group), even when that action is not free. The gist is that it is "complicated" and resembles the theory of a "bad group" of finite Morley rank. torus.math.uiuc.edu /cal/math/cal?...&regexp=Group+Theory&use=Find   (1090 words)

Research A countable group is amenable if and only if up to OE it gives rise to exactly one countable ergodic measure preserving equivalence relation on a standard Borel probability space. A lcsc group which is non-amenable and the product of two non-compact lcsc groups gives rise to a non-treeable orbit equivalence relation whenever it acts freely by measure preserving transformations on a standard Borel probability space. Given a non-trivial group of isometries on a space X we can find a bounded uniformly continuous function from X which continues to obtain high oscillation on the image of X under any isometric embedding arising in the closure of the group. www.math.ucla.edu /~greg/research.html   (1625 words)

Harry's Dive Shop Holy week, the week before Easter Sunday of 2003, a small group of divers went to Curacao to escape the horrible winter temperatures of the Deep South. The group did a few boat dives but the majority of the dives were from the dock, right out in front of the resort. With ideal conditions and an amenable group of divers, the world was theirs to do with as they saw fit. www.harrysdiveshop.com /logbook/logs/curacao03.htm   (186 words)

AMCA: The von Neumann algebra of the canonical equivalence relation of the Thompson group by Gabriel Picioroaga   (Site not responding. Last check: 2007-10-13) We study the equivalence relation R generated by the (non-free) action of the Thompson group F on the unit interval. It is known that Thompson's group has cost 1, so that studying equivalence relations generated by F could be useful in attacking the problem of (non)amenability of F. We prove that R above is treeable. However, we think the treeability of R together with the hyperfiniteness of M(R) bring a flavor of amenability on the Thompson group. at.yorku.ca /c/a/n/e/24.htm   (296 words)

N. Higson Abstract   (Site not responding. Last check: 2007-10-13) It has recently been shown that every countable amenable group admits a "metrically proper" isometric action on an infinite dimensional Euclidean space. Several other classes of groups also so act, including free groups and proper groups of isometries of hyperbolic space. The conjecture, which belongs to C*-algebra theory, has some important consequences in topology (where the isometry groups in question appear as fundamental groups of manifolds) and at the present time some of these results are accessible only by our C*-algebraic techniques. www.math.uiuc.edu /Colloquia/97FA/higson.html   (160 words)

[No title]   (Site not responding. Last check: 2007-10-13) It may be argued that this technically difficult area has thus far constituted the deepest aspect of the subject. Takesaki, together with several coworkers, has very recently completed one of the central classification programs for discrete amenable group actions on operator algebras. He also plans to work once again on his earlier theory of duality, which is currently playing a major role in quantum group theory. www.cs.utexas.edu /users/yguan/NSFAbstracts/Abstracts/MPS/DMS.MPS.a9500882.txt   (289 words)

PlanetMath: A finitely generated group has only finitely many subgroups of a given index owned by avf A groups embeds into its profinite completion if and only if it is residually finite owned by avf alternating group is a normal subgroup of the symmetric group owned by mathcam planetmath.org /encyclopedia/A   (1878 words)

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