Non-vanishing Results for Higher L^2-Betti Numbers of Locally Compact Groups and S-arithmetic Lattices

Details Non-vanishing Results for Higher L^2-Betti Numbers of Locally Compact Groups and S-arithmetic Lattices Non-vanishing Results for Higher L^2-Betti Numbers of Locally Compact Groups and S-arithmetic Lattices

2011-04-17

arxiv.org/abs/1104.3294v1

Mathematics

Group Theory

49 pages

Scientific paper

We set up a general machinery to define a notion of $L^2$-Betti numbers for 2nd countable, unimodular, locally compact groups in the spirit of L{\"u}ck's approach to $L^2$-invariants for countable discrete groups. These $L^2$-Betti numbers of a locally compact group $G$ are related to those of lattices in $G$, and for totally disconnected $G$ the theory bears resemblance to that for countable discrete groups, and coincides in dimension one with a prior definition of first $L^2$-Betti number for locally finite graphs by Gaboriau. Our main application applies a spectral sequence argument to get lower bounds for $L^2$-Betti numbers of lattices in a direct product $G=G_1\times G_2$.

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