Lück's approximation theorem

The approximation of Luck is a theorem from the mathematical field of algebraic topology . It relates the L ² -Betti numbers of a space to the usual Betti numbers of its finite overlays . ${\displaystyle b_{k}^{(2)}(X)}$ ${\displaystyle X}$ ${\displaystyle b_{k}(X_{i})}$ ${\displaystyle X_{i}}$

Statement of the sentence

Let be a finite CW-complex with a residual finite fundamental group . Because of the residual finiteness, there is a descending chain of normal parts with and . Let be the overlay of with cover group . Then ${\displaystyle X}$ ${\displaystyle \Gamma }$ ${\displaystyle \left[\Gamma \colon \Gamma _{i}\right]<\infty }$ ${\displaystyle \bigcap _{i}\Gamma _{i}=0}$ ${\displaystyle X_{i}}$ ${\displaystyle X}$ ${\displaystyle \Gamma _{i}}$

${\displaystyle b_{k}^{(2)}(X)=\lim _{i\to \infty }{\frac {b_{k}(X_{i})}{\left[\Gamma \colon \Gamma _{i}\right]}}.}$

Let in particular be a finitely presented , residual finite group and a descending chain of normal subdivisions with and , then is ${\displaystyle \Gamma }$ ${\displaystyle \Gamma =\Gamma _{0}\supset \Gamma _{1}\supset \Gamma _{2}\supset \ldots }$ ${\displaystyle \left[\Gamma \colon \Gamma _{i}\right]<\infty }$ ${\displaystyle \bigcap _{i}\Gamma _{i}=0}$

${\displaystyle b_{k}^{(2)}(\Gamma )=\lim _{i\to \infty }{\frac {b_{k}(\Gamma _{i})}{\left[\Gamma \colon \Gamma _{i}\right]}}.}$

The approximation theorem also applies to homology with coefficients in any field with the characteristic zero.

Generalization for grids in symmetrical spaces

Let be a symmetric space of non-compact type and a uniformly discrete sequence of co-compact grids in , for which converges to Benjamini-Schramm . Then ${\displaystyle X}$ ${\displaystyle \Gamma _{n}}$ ${\displaystyle X}$ ${\displaystyle \Gamma _{n}\backslash X}$ ${\displaystyle X}$

${\displaystyle \lim _{i\to \infty }{\frac {b_{k}(\Gamma _{i})}{vol(\Gamma _{i}\backslash X)}}=\beta _{k}^{(2)}(X)}$

With

${\displaystyle \beta _{k}^{(2)}(X)=0}$ For ${\displaystyle k\not ={\frac {1}{2}}dim(X)}$

and

${\displaystyle \beta _{{\frac {1}{2}}dim(X)}^{(2)}(X)={\frac {\chi (X^{d})}{vol(X^{d})}}}$

for the too dual compact symmetrical space. ${\displaystyle X}$

literature

• Wolfgang Lück : Approximating L2-invariants by their finite-dimensional analogues. GAFA 4 (1994), pp. 458-490.
• Pierre Pansu : Introduction to L2 -Betti numbers.
• Michail Gromov : Asymptotic Invariants of Infinite Groups. (Chapter 8)
• Wolfgang Lück: L2-Invariants: Theory and Applications to Geometry and K-Theory.

Individual evidence

1. ↑ Miklos Abert, Nicolas Bergeron, Ian Biringer, Tsachik Gelander, Nikolay Nikolov, Jean Raimbault, Iddo Samet: On the growth of L ² -invariants for sequences of lattices in Lie groups . Ann. Math. 185 (2017), pp. 711-790