Property A and coarse embedding for locally compact groups

In the study of the Novikov conjecture, property A and coarse embedding of metric spaces were introduced by Yu and Gromov, respectively. The main topic of the thesis is property A and coarse embedding for locally compact second countable groups. We prove that many of the results that are known to hold in the discrete setting, hold also in the locally compact setting.

In a joint work with Deprez, we show that property A is equivalent to amenability at infinity and the strong Novikov conjecture is true for every locally compact group that embeds coarsely into a Hilbert space (see Article A).

In a joint work with Deprez, we show a number of permanence properties of property A and coarse embeddability into Hilbert spaces (see section 4).

In section 6 we give a completely bounded Schur multiplier characterization of locally compact groups with property A. In particular, weakly amenable groups have property A.

In a joint work with Knudby, we characterize the connected simple Lie groups with the discrete topology that have different approximation properties (see Article B). Moreover, we give a contractive Schur multiplier characterization of locally compact groups coarsely embeddable into Hilbert spaces (see Article C). Consequently, all locally compact groups whose weak Haagerup constant is 1 embed coarsely into Hilbert spaces.

In a joint work with Brodzki and Cave, we show that exactness of a locally compact second countable group is equivalent to amenability at infinity, which solves an open problem raised by Anantharaman-Delaroche (see section 8).

Supervisor: Prof. Ryszard Nest, Math, University of Copenhagen

Co-supervisor: Prof. Uffe Haagerup, Math, University of Copenhagen

Assessment committee:

Ass. Prof. Magdalena Musat (chairman), MATH, University of Copenhagen

Prof. Claire Anantaraman-Delaroche, Université d’Orléans

Prof. Alain Valette, Université de Neuchâtel