Ph.D Thesis | |

Ph.D Student | Lavy Omer |
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Subject | Groups Actions on Banach Spaces |

Department | Department of Mathematics |

Supervisor | ASSOCIATE PROF. Uri Bader |

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.