This work deals with several aspects concerning with group actions. The first part deals with isometric action of the Steinberg group defined over the non-commutative ring of polynomials over finite field.
We show that such groups cannot act isometrically on Hadamard manifolds. When manifolds of infinite dimension are taken into account we show that under the assumption that the manifold is pinched (i.e. its sectional curvature is bounded from below as well) every isometric action has a fixed point.
The second part deals with group actions on l_p spaces. Given a locally compact group G, we will be interested in describing the rational numbers, p for which G has property F_lp. We show that this set (whenever not empty) is a section (possibly pinched at 2). This is a summary of a joint work with Baptiste Olivier.
In the third part we consider the group generated by elementary matrices, G=EL_n(Z[t]).This group has a natural action on the dual of Z[t]^n. We are interested in characterization of ergodic measures on the dual of Z[t]^n which are invariant under this action. We show that such measures are (convex combination of) Haar measures defined on (maybe cosets of) invariant subgroups of the dual. This section is a summary of an ongoing project with Arie Levit.