Approximating L^2 invariants by finite-dimensional analogues.

2006-11-22 22:42:51 +0000 talks

I gave a couple of seminar talks on

MR1280122

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 on this subject,

MR1926649

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