<!DOCTYPE html>
<html>
<head>
<title>Set Theory and Analysis</title>
</head>
<body>
<p>
Tuesday, 10 June 2025 - 10:00 to 11:30 <br />
Place: IM, konÃrna
</p>
<p>
Speaker: Manfred Droste, University of Leipzig<br />
Title: Permutation groups, groups of measure-preserving or ergodic transformations and the Bergman property
</p>
<p class="ql-ed">
Abstract <br />
<p>We will show that the group of all measure-preserving permutations of the unit interval and the full group of an ergodic transformation of the unit interval have the Bergman property. Here, a group $G$ is said to have the Bergman property, if for any generating subset $E$ of $G$, already some bounded power of $E \cup E^{-1} \cup {1}$ covers $G$. This property arose in a recent interesting paper of Bergman where it was derived for the infinite symmetric groups. We give a general sufficient criterion for permutation groups $G$ to have the Bergman property. We show that the criterion applies to a range of further groups, including</p><p> the homeomorphism groups of the rationals, the irrationals, or Cantor's set, the groups of all measure preserving or non-singular transformations of the reals, and to sufficiently transitive groups of ergodic transformations of the reals.</p>
</p>
<p>
For more information see the seminar web page at <br />
https://www.math.cas.cz/index.php/events/seminar/6
</p>
<p>
Set Theory and Analysis mailing list <br />
settfa@math.cas.cz <br />
https://list.math.cas.cz/listinfo/settfa@math.cas.cz
</p>
</body>
</html>