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Bounded cohomology of group was introduced in a geometrical and topological context by Gromov in the beginning of the 80’s. Shortly after, Burger and Monod extended the definition to the continuous bounded cohomology of topological groups. Despite being a seemingly harmless modification of the classical definition of group cohomology, bounded cohomology has proved extremely difficult to compute. For example for the free group it is known only up to degree 3 while for SL_2® only up to degree 4. 
We establish the first complete computation of the continuous bounded cohomology of a simple algebraic group by proving that the continuous bounded cohomology of SL_2(k) vanishes in all positive degrees for any non-Archimedean local field k. We will present the proof which is based on a simple geometrical argument. 
Joint work with Nicolas Monod.

This talk is part of the Isaac Newton Institute Seminar Series series.

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