Workshop on Invariant Random Subgroups

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<h4>Presentation</h4> <div class="level4"> <p> The Neretin group of a regular tree is defined as a group of "piecewise automorphisms" on the boundary. It can be endowed with a locally compact topology, and has no lattices. It is a natural question to ask whether it has IRSs beyond the trivial ones. It is likely that a proof that IRSs "parabolic" subgroups of the Neretin groups (subgroups fixing a point on the boundary of the tree) can be classified using a modification of an argument for similar groups in the discrete case, and we hope that this might help proving that there are nontrivial IRSs in the Neretin groups. </p> </div> <h4>Lecture plan</h4> <div class="level4"> <ul> <li class="level1"> The Neretin group, and why it has no lattices; </li> <li class="level1"> Groups acting on trees which have no lattices; </li> <li class="level1"> "Parabolic" subgroups and their IRSs: a continuous version of a result of Thomas–Tucker-Drob? </li> </ul> </div> <h4>References</h4> <div class="level4"> <ul> <li class="level1"> Łukasz Garncarek, Nir Lazarovich, <em>The Neretin groups</em>, <a title="https://arxiv.org/abs/1502.00991" rel="nofollow" class="urlextern" href="https://arxiv.org/abs/1502.00991">Arxiv version</a>. </li> <li class="level1"> Uri Bader, Pierre-Emmanuel Caprace, Tsachik Gelander, Shahar Mozes, <em>Simple groups without lattices</em>, Bull. LMS 2012, <a title="https://arxiv.org/abs/1008.2911" rel="nofollow" class="urlextern" href="https://arxiv.org/abs/1008.2911">Arxiv version</a>. </li> <li class="level1"> Adrien le Boudec, <em>Groups acting on trees with almost prescribed local action</em>, Comm. Math. Helv. 2016, <a title="https://arxiv.org/abs/1505.01363" rel="nofollow" class="urlextern" href="https://arxiv.org/abs/1505.01363">Arxiv version</a></li> <li class="level1"> Simon Thomas, Robin Tucker-Drob, <em>Invariant random subgroups of strictly diagonal limits of finite symmetric groups</em>, Bull. LMS 2014, <a title="https://arxiv.org/abs/1402.4837" rel="nofollow" class="urlextern" href="https://arxiv.org/abs/1402.4837">Arxiv version</a>. </li> </ul> </div>

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