# Seminar: "About some strange actions of the free group recently found by A. Ol'shanskii" (Enric Ventura, UPC)

Date: Thursday April 10th, 16-17.

Place: CRM, room C1/028

Títle: "About some strange actions of the free group recently found by A. Ol'shanskii".

Speaker: Enric Ventura, UPC.

Abstract: In this talk I'll explain a recent preprint by A. Olshanskii, where he proves the existence of the following kind of "strange" actions: there exists an action of the free group $F_r$, $r\geq 2$, on an infinite set $S$ simultaneously satisfying the following properties: (1) it is highly transitive (i.e., for every $k\geq 1$ there is only one orbit of $k$-tuples of points); (2) it is faithful (i.e., only the trivial element fixes everything); and (3) its restriction to ANY finitely generated subgroup of infinite index is LOCALLY FINITE (i.e., all orbits are finite).

These actions are quite strange because property (3) is strongly pointing (intuitively) into the opposite direction as properties (1) and (2). The proof is not difficult, just playing with ability in the lattice of subgroups of the free group, and combining both algebraic and graphical techniques.