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Title: Perfect kernel of generalized Baumslag-Solitar groups

Abstract: Endowed with the Chabauty topology, the space of subgroups Sub(G) of any infinite countable group G is a closed subset of the Cantor space, on which G acts by conjugation. The perfect kernel of G is the largest closed subset of Sub(G) without isolated points. It is invariant by conjugation.
In this talk, we will see how the action of a group G on an oriented tree T can give information on the perfect kernel of G, and on the dynamics induced by the action by conjugation on it. In 2023, Azuelos and Gaboriau studied the case where the stabilizers "vanish", i.e. there exists an edge path of T whose stabilizer is finite. They proved that the closure of the G-invariant subset Sub|•\T|∞(G), which consists in the set of subgroups of G acting on T with infinitely many orbits of edges, is included in the perfect kernel and contains a dense orbit.
Generalizing results obtained by Carderi, Gaboriau, Le Maître and Stalder, we will study the space of subgroups of generalized Baumslag-Solitar groups, i.e. groups acting cocompactly on an oriented tree with infinite cyclic edge and vertex stabilizers. These are typical examples of non vanishing stabilizers.
After proving that the perfect kernel of such non amenable group exactly consists in Sub|•\T|∞(G), we show that this leads to very different dynamics on it. In particular, we show the existence of an infinite countable G-invariant partition of the perfect kernel such that:
- one piece of the decomposition is closed, and all the other ones are open (and closed iff G is virtually the direct product of a free group with ℤ);
- there exists a dense orbit in each of these pieces.

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