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Invariants and Boundaries of Spaces

The project aims to study invariants and geometric objects arising in the context of spaces of non-positive curvature.  It focuses on two main themes.

(1) Developing new techniques for the explicit computation of the bounded cohomology of Lie groups and the simplicial volume of locally symmetric spaces.  The methods are from analysis, geometry and homological algebra, involving partial differential equations, the geometry of Lie groups, and polylogarithm functions.

(2) Investigating the interplay between gauge-theoretic and geometric compactifications of the moduli space of representations of surface groups into Lie groups.  The methods are from global analysis and hyperbolic geometry, involving Higgs bundles, harmonic maps, and pleated surfaces.


Publications

We develop an algebro-analytic framework for the systematic study of the continuous bounded cohomology of Lie groups in large degree. As an application, we examine the continuous bounded cohomology of PSL(2,R) with trivial real coefficients in all degrees greater than two. We prove a vanishing result for strongly reducible classes, thus providing further evidence for a conjecture of Monod. On the cochain level, our method yields explicit formulas for cohomological primitives of arbitrary bounded cocycles.

 

Related project(s):
27Invariants and boundaries of spaces

We present a new technique that employs partial differential equations in order to explicitly construct primitives in the continuous bounded cohomology of Lie groups. As an application, we prove a vanishing theorem for the continuous bounded cohomology of SL(2,R) in degree 4, establishing a special case of a conjecture of Monod.

 

JournalGeometry & Topology
Volume19
Pages3603–3643
Link to preprint version
Link to published version

Related project(s):
27Invariants and boundaries of spaces

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Team Members

Dr. Andreas Ott
Project leader
Ruprecht-Karls-Universität Heidelberg
aott(at)mathi.uni-heidelberg.de

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