Amenability versus the ideal of wandering sets - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T04:58:56Z http://mathoverflow.net/feeds/question/45781 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/45781/amenability-versus-the-ideal-of-wandering-sets Amenability versus the ideal of wandering sets Justin Moore 2010-11-12T03:33:39Z 2010-11-12T14:57:14Z <p>Let $G$ be a finitely generated group acting on a set $S$ (on the right). Define the heirarchy of "marginal sets" as follows:</p> <p><li> The emptyset is 0-marginal.</p> <p><li> A set E is $(k+1)$-marginal if $E$ can be covered by finitely many sets $A$ such that for some $k$-marginal set $B$ and some $g$ in $G$, if $a$ is in $A$ and $a \cdot g^i$ is in $A$ for some $i > 0$, then there is a $j &lt; i$ such that $a \cdot g^j$ is in $B$.</p> <p><li> a set is marginal if it is $k$-marginal for some $k$.</p> <p>So if we recall that $A$ is wandering if, for some $g$ in $G$, the sets $A \cdot g^i$ $(i &lt; \infty)$ are pairwise disjoint, then 1-marginal just means that the set is a finite union of wandering sets.</p> <p>The point is that marginal sets are assigned measure 0 by any invariant measure (or even by any invariant exhaustive submeasure --- see my other recent question).</p> <p>Now for the question(s):</p> <p><li> Are there non-amenable actions which are aperiodic but not marginal? (Notice that if $G$ is a non-amenable torsion group acting on itself then the emptyset is the only marginal set.)</p> <p><li> Can $S$ be $(k+1)$-marginal but not $k$-marginal for some $k > 0$?</p> <p><li> Has the notion of marginality been considered (and given a name)? (I used the notion to give estimates in my proof of a lower bound for the Folner function for Thompson's $F$, but it seems this "must" have been considered before.)</p>