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.

• The Neretin group, and why it has no lattices;
• Groups acting on trees which have no lattices;
• “Parabolic” subgroups and their IRSs: a continuous version of a result of Thomas–Tucker-Drob?

• Łukasz Garncarek, Nir Lazarovich, The Neretin groups, Arxiv version.
• Uri Bader, Pierre-Emmanuel Caprace, Tsachik Gelander, Shahar Mozes, Simple groups without lattices, Bull. LMS 2012, Arxiv version.
• Adrien le Boudec, Groups acting on trees with almost prescribed local action, Comm. Math. Helv. 2016, Arxiv version
• Simon Thomas, Robin Tucker-Drob, Invariant random subgroups of strictly diagonal limits of finite symmetric groups, Bull. LMS 2014, Arxiv version.