Borel invariant of measurable cocycles of 3-manifold groups (Alessio Savini, University of Bologna)

05.11.2019 10:30

A quite useful philosophy in mathematics is to use the sharpness of an inequality regarding the "shape" of a topological space X in order to detect a precise geometry: more precisely the maximal value of the inequality usually allows to identify a specific geometric structure on X. For instance one can think either to the arithmetic/geometric mean inequality or to the isoperimetric inequality.
Something similar happens in the world of Zimmer's cocycles theory. In this seminar we are going to focus our attention on PSL(n,C)-valued measurable cocycles associated to a torsion-free lattice G of PSL(2,C). If such a measurable cocycle admits a boundary map, one can define a numerical invariant called Borel invariant, which well behaves along the associated PSL(n,C)-cohomology class. Additionally the absolute value of this numerical invariant is bounded by a constant times the covolume of G. Surprisingly the maximal value is attained if and only if the cocycle is cohomologous to the composition of the standard lattice embedding with the irreducible representation of PSL(2,C) into PSL(n,C).

Lieu

Room 623, Séminaire "Groupes et Géométrie"

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