Rokhlin lemma for arbitrary infinite groups. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T04:02:39Z http://mathoverflow.net/feeds/question/80579 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/80579/rokhlin-lemma-for-arbitrary-infinite-groups Rokhlin lemma for arbitrary infinite groups. Łukasz Grabowski 2011-11-10T11:13:23Z 2011-11-14T22:30:43Z <p>Let $G$ be an at most countable discrete group acting freely on a standard probability measure space $X$ in a measure preserving way.</p> <p>It is well known that if $G$ is a finite group then this action admits a fundamental domain. As pointed out by Andreas below, by Rokhlin lemma, if $G$ contains an element of infinite order we can find an $(\varepsilon, N)$-fundamentalish domain $U$, where the latter is defined as follows:</p> <p>Call a set $U\subset X$ an $(\varepsilon, N)$-fundamentalish domain iff there exist $N$ elements $g_1, \ldots, g_N$ of $G$ such that the sets $g_i(U)$ are pairwise disjoint and the measure of their union is at least $1-\varepsilon$. </p> <blockquote> <p><strong>Question:</strong> If $G$ is an infinite group, $N_0$ is a natural number, $\varepsilon_0$ is a positive real number, does there exist an $(\varepsilon, N)$-fundamentalish domain with $\varepsilon&lt;\varepsilon_0$ and $N>N_0$?</p> </blockquote> <p>For example when the action is profinite and "transitive on each level", then clearly answer is positive: there exist $(0,N)$-fundamentalish domains for arbitrary large $N$.</p> http://mathoverflow.net/questions/80579/rokhlin-lemma-for-arbitrary-infinite-groups/80934#80934 Answer by Brandon Seward for Rokhlin lemma for arbitrary infinite groups. Brandon Seward 2011-11-14T22:03:46Z 2011-11-14T22:30:43Z <p>If you fix $N$ and group elements $g_1$, ..., $g_N \in G$, then your question becomes closely related to tilings of groups. Specifically, in Chapter 2, section 2 of "<a href="http://www.springerlink.com/content/7p0r815012335131/" rel="nofollow">Entropy and Isomorphism Theorems for Actions of Amenable Groups</a>," Ornstein and Weiss prove:</p> <p>Let $G$ be a countable group acting freely and measure preservingly on a standard probability space $(X, \mu)$. Fix a finite set $T \subseteq G$. If for every $\epsilon > 0$ there is a measurable set $U \subseteq X$ such that the $T$-translates of $U$ are disjoint and $\mu(T \cdot U) > 1 - \epsilon$, then $T$ tiles $G$ in the sense that there is a set of centers $C \subseteq G$ such that the sets $Tc$ ($c \in C$) partition $G$.</p> <p>They also prove that if $G$ is amenable then the converse holds. So if $G$ is amenable and $T$ is a tile for $G$, then for every free probability measure preserving action of $G$ and every $\epsilon > 0$ there is a $(\epsilon, |T|)$-fundamentalish domain. Thus a natural question is: which amenable groups admit arbitrarily large finite tiles?</p> <p>Weiss called a group $G$ MT (mono-tileable) if for every finite set $F \subseteq G$ there is a finite tile $T \subseteq G$ containing $F$. In "<a href="http://books.google.com/books?id=NDJ0rRuSMScC&amp;pg=PA257&amp;lpg=PA257&amp;dq=monotileable+amenable+groups&amp;source=bl&amp;ots=DpqekXyi_H&amp;sig=ShnrG6LH5nqIqT1QKGWVcxJFXlM&amp;hl=en&amp;ei=aovBTs3_D8W02wWAs-GxBQ&amp;sa=X&amp;oi=book_result&amp;ct=result&amp;resnum=1&amp;ved=0CBoQ6AEwAA#v=onepage&amp;q=monotileable%2520amenable%2520groups&amp;f=false" rel="nofollow">Monotileable amenable groups</a>," Weiss proved that all solvable groups and all residually finite groups are MT. In "<a href="http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.ijm/1256047608" rel="nofollow">Elementary Amenable Groups</a>," Chou proved that all elementary amenable groups and all free products of non-trivial groups are MT. So in particular, your question has a positive answer whenever the group $G$ is elementary amenable. A stronger tiling condition, called ccc, is studied in chapter 4 of "<a href="http://www-personal.umich.edu/~bseward/Files/Group%20Colorings%20and%20Bernoulli%20Subflows.pdf" rel="nofollow">Groups Colorings and Bernoulli Subflows</a>" (this paper is in preparation). Weaker properties of poly-MT and poly-ccc are studied in "<a href="http://www-personal.umich.edu/~bseward/Files/Burnside%27s%20Problem,%20Spanning%20Trees,%20and%20Tilings.pdf" rel="nofollow">Burnside's Problem, spanning trees, and tilings</a>." To the best of my knowledge these are the only papers which study tilings of countable groups.</p>