Approximating L^2 invariants by finite-dimensional analogues.

 November 22, 2006 talks

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I gave a couple of seminar talks on [1].

Here’s the main result in the paper. Let X be a CW-complex, and filter \Gamma = \pi_1 X as \Gamma = \Gamma_1 \rhd \Gamma_2 \rhd \cdots with [\Gamma_i : \Gamma_{i+1}] < \infty so that \bigcap_i \Gamma_i = \{ 1 \} . Let X_i be the cover of X corresponding to the normal subgroup \Gamma_i .

Then, the limit of the “normalized” Betti numbers \lim_{j \to \infty} b_j( X_i ) / [\Gamma : \Gamma_i] is equal to b^{(2)}_j(X) , the L^2 Betti number of X . In particular, the limit of the normalized Betti numbers is independent of the filtration! In other words, we have “approximated” the L^2 invariant by a limit of finite-dimensional approximations.

The awesome thing about this result is how “easy” the proof is; it’s just some linear algebra (eh, functional analysis), but I don’t claim to have a very conceptual understanding of why it is true. In the big book [2] on this subject, there is a more conceptual explanation of the proof; the book also mentions some basic generalizations.

[1] W. Lück, Approximating L\sp 2 -invariants by their finite-dimensional analogues, Geom. Funct. Anal. 4 (1994) 455–481.

[2] W. Lück, L\sp 2 -invariants: Theory and applications to geometry and K -theory, Springer-Verlag, Berlin, 2002.