Mathematisches Institut, Sidlerstrasse 5, CH-3012 Bern Philosophischnaturwissenschaftliche Fakultät Departement Mathematik und Statistik Mathematisches Institut Mathematical Colloquia ____________________________________________________________________________ Monday, 17 October 2016 17:15 h, Lecture Room B 78 Prof. Dr. Masato Mimura, Tohoku University / EPFL Bounded and unbounded generation; and fixed point properties Abstract: Fix a complete metric space. We say that a finitely generated group has the fixed point property (f.p.p.) relative to it if every isometric action admits a global fixed point. A surprising theorem of Kazhdan implies that SL(n,Z), the special linear group over the ring of integers, has f.p.p. relative to Euclidean spaces, and even to Hilbert spaces (that is, infinite dimensional Euclidean spaces), provided that n is at least 3. One influential way to show that f.p.p. was invented by Y. Shalom [Publ. IHES, 1999]. It employs so-called "Bounded Generation" (BG) of a group by finite family of subgroups. This method is completely powerful as long as BG is available. However, in general, BG condition is too strong to expect. In this talk, I will present a recent resolution to the barrier of the case where BG is invalid/unknown. In this talk, I will give the definition of BG; exhibit some examples; and explain why this was essential in previous works. Finally, I will describe our new approach to f.p.p., that is free from exploiting BG. Sekretariat, Mathematisches Institut, Sidlerstrasse 5, CH-3012 Bern, Tel. +41 (0)31 631 88 21, Fax +41 (0)31 631 85 10 [email protected], www.math.unibe.ch