Strong approximation in random towers of graphs
| dc.creator | Glasner, Yair | |
| dc.date | 2009-05-01 | |
| dc.date.accessioned | 2026-07-07T13:11:14Z | |
| dc.date.available | 2026-07-07T13:11:14Z | |
| dc.description | The term "strong approximation" is used to describe phenomena where an arithmetic group as well as all of its Zariski dense subgroups have a large image in the congruence quotients. We exhibit analogues of such phenomena in a probabilistic, rather than arithmetic, setting. Let T be the binary rooted tree, Aut(T) its automorphism group. To a given m-tuple a = {a_1,a_2,...,a_m} in Aut(T), we associate a tower of 2m-regular Schreier graphs ...X_n-->X_{n-1}-->...-->X_0. The vertices of X_n are the n^{th} level of the tree and two such are connected by an edge if a generator takes one to the other. When {a_i} are independent Haar-random elements of Aut(T) we retrieve the standard model for iterated random 2-lifts studied, for example by Bilu-Linial. If w={w_1,w_2,...,w_l} are words in the free group F_m, the random substitutions w(a) := {w_1(a),...,w_l(a)} give rise to new models for random towers of 2l-regular graphs: ...Y_n-->Y_{n-1}-->...-->Y_0. With the above notation, the following hold almost surely, for every non cyclic subgroup D in F_m: (i) the graphs $Y_n$ have a bounded number of connected components, (ii) these connected components form a family of expander graphs, (iii) the closure of D has positive Hausdorff dimension as a subgroup of the (metric) group Aut(T). | |
| dc.description | 30 pages, 3 figures | |
| dc.identifier | https://arxiv.org/abs/0905.0156 | |
| dc.identifier | http://arxiv.org/abs/0905.0156 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/229225 | |
| dc.subject | Combinatorics | |
| dc.subject | Group Theory | |
| dc.subject | 20E08, 60J80 (Primary); 20B27, 28A78, 11F06 (Secondary) | |
| dc.title | Strong approximation in random towers of graphs | |
| dc.type | text |