k is one cat » Approximating L^2 invariants by finite-dimensional analogues.

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Approximating L^2 invariants by finite-dimensional analogues.

I gave a couple of seminar talks on

Lück, W. Approximating $L\sp 2$ -invariants by their finite-dimensional analogues. Geom. Funct. Anal. 4 (1994), no. 4, 455—481.

Here’s the main result in the paper. Let 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 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 . 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 on this subject,

Lück, Wolfgang $L\sp 2$ -invariants: theory and applications to geometry and -theory.Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 44. Springer-Verlag, Berlin, 2002. xvi+595 pp. ISBN: 3-540-43566-2

there is a more conceptual explanation of the proof; the book also mentions some basic generalizations.

Posted: November 22nd, 2006 under Talks.
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Approximating L^2 invariants by finite-dimensional analogues.

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