Benjamini-Schramm Convergence

In mathematics , Benjamini-Schramm convergence or BS convergence for short is a term originally derived from graph theory and now also used in geometry and topology .

The idea is to approximate infinite graphs or non-compact Riemannian manifolds by finite graphs or compact Riemannian manifold .

Benjamini-Schramm convergence of graphs

The following definition was introduced into graph theory by Itai Benjamini and Oded Schramm .

definition

For each graph we consider the probability measure on the set of root graphs , which corresponds to the uniform distribution on the set of root graphs for nodes of . (In particular, it has its support on the set of root graphs whose underlying graph is.) ${\displaystyle G}$ ${\displaystyle \mu _{G}}$ ${\displaystyle (G,o)}$ ${\displaystyle o}$ ${\displaystyle G}$ ${\displaystyle \mu _{G}}$ ${\displaystyle G}$

On a graph can be a metric defined by the fact that each edge is assigned to the length of the first For a root graph and denotes the subgraph that is spanned by all nodes with a distance of less than . ${\displaystyle G}$ ${\displaystyle (G,o)}$ ${\displaystyle r\in \mathbb {N} }$ ${\displaystyle B_{G}(o,r)}$ ${\displaystyle o}$ ${\displaystyle r}$

A sequence of graphs of bounded valence BS-converges to a graph if for each root graph and each the probability that it is too isomorphic converges to the probability that it is too isomorphic. ${\displaystyle G_{n}}$ ${\displaystyle G_{\infty }}$ ${\displaystyle (H,o_{H})}$ ${\displaystyle r\in \mathbb {N} }$ ${\displaystyle (G_{n},o)}$ ${\displaystyle (H,o_{H})}$ ${\displaystyle (G_{\infty },o)}$ ${\displaystyle (H,o_{H})}$

The circle graph , , and ${\displaystyle C_{3}}$ ${\displaystyle C_{4}}$ ${\displaystyle C_{5}}$ ${\displaystyle C_{6}}$

example

The sequence of the circle graphs BS converges to the Cayley graph of the group of integers , that is to say the infinite linear graph . ${\displaystyle C_{n}}$ ${\displaystyle P_{\infty }}$

Benjamini-Schramm convergence of Riemann manifolds

definition

We provide the set of dotted Riemannian manifolds with the Gromov-Hausdorff topology . ${\displaystyle {\mathcal {M}}}$

Let be a (non-compact) Riemannian manifold and a sequence of lattices in the isometric group . ${\displaystyle M}$ ${\displaystyle \Gamma _{n}}$ ${\displaystyle Isom(M)}$

One says that the sequence of Riemannian manifolds converges against in the sense of Benjamini-Schramm if for all the probability that the sphere of radius around a random point in is isometric to the corresponding sphere of radius in converges for against . ${\displaystyle M_{n}=\Gamma _{n}/M}$ ${\displaystyle M}$ ${\displaystyle R>0}$ ${\displaystyle R}$ ${\displaystyle M_{n}}$ ${\displaystyle R}$ ${\displaystyle M}$ ${\displaystyle n\to \infty }$ ${\displaystyle 1}$

An equivalent condition is that for each ${\displaystyle R>0}$

${\displaystyle \lim _{n\to \infty }{\frac {vol((M_{n})_{<R})}{vol(M_{n})}}=0}$

holds, where the - thin part of or the Injektivitätsradius designated. ${\displaystyle (M_{n})_{<R}=\left\{x\in M_{n}\colon inj_{M_{n}}(x)<R\right\}}$ ${\displaystyle R}$ ${\displaystyle M_{n}}$ ${\displaystyle inj_{M_{n}}}$

example

Let be a co-compact lattice and a sequence of normal subgroups with . Then for sufficiently large , i.e. BS-converges, the consequence is to . ${\displaystyle \Gamma \subset Isom(M)}$ ${\displaystyle \Gamma _{n}\subset \Gamma }$ ${\displaystyle \textstyle \bigcap _{n}\Gamma _{n}=\left\{0\right\}}$ ${\displaystyle (M_{n})_{<R}=\emptyset }$ ${\displaystyle n}$ ${\displaystyle M_{n}=\Gamma _{n}/M}$ ${\displaystyle M}$

Benjamini-Schramm convergence of metric spaces

The following general definition includes the previous two.

We provide the set of dotted , actual , compact metric spaces with the Gromov-Hausdorff topology. ${\displaystyle {\mathcal {X}}}$

Let be an actual, compact, metric space with a probability measure . This defines a probability measure on-called probability that the distribution of the according to the amount of the dotted areas with equal. (In particular, it has its bearer on the set of dotted metric spaces whose underlying metric space is.) ${\displaystyle X}$ ${\displaystyle \mu }$ ${\displaystyle \mu _{X}}$ ${\displaystyle {\mathcal {X}}}$ ${\displaystyle x\in X}$ ${\displaystyle \mu }$ ${\displaystyle (X,x)\in >{\mathcal {X}}}$ ${\displaystyle x\in X}$ ${\displaystyle \mu _{X}}$ ${\displaystyle X}$

Let be a sequence of actual, compact, metric spaces with a probability measure . It is said that the result Benjamini-Schramm-converges if the sequence in the weak - * - topology against a measure to converge. ${\displaystyle X_{n}}$ ${\displaystyle \mu _{n}}$ ${\displaystyle X_{n}}$ ${\displaystyle \mu _{X_{n}}}$ ${\displaystyle \mu _{\infty }}$ ${\displaystyle {\mathcal {X}}}$

Individual evidence

1. ^ Benjamini, Schramm: Recurrence of distributional limits of finite planar graphs. Electron. J. Probab. 6 (2001), No. 23, Project Euclid
2. ↑ Abert, Bergeron, Biringer, Gelander, Nikolov, Raimbault, Samet: On the growth of L ² -invariants for sequences of lattices in Lie groups. Ann. of Math. (2) 185 (2017), 711-790.