图的随机塔中的强逼近
Strong approximation in random towers of graphs
摘要：
术语“强逼近”用来描述一个算术群及其所有的Zebraki稠密子群在同余式中有一个大像的现象。我们展示类似的现象，在概率，而不是算术，设置。设T是二元有根树，Aut（T）是它的自同构群.对于给定的m元组a = {a_1，a_2，.，在Aut（T）中，我们将一个2 m-正则Schreier图的塔. X_n--> X_{n-1} -->. --> X_0。X_n的顶点是树的第n层，如果生成器将一个顶点取到另一个顶点，则两个顶点由边连接。当{a_i}是Aut（T）的独立Haar-随机元时，我们恢复了Bilu-Linial研究的迭代随机2-提升的标准模型.如果w={w_1，w_2，.，w_l}是自由群F_m中的字，随机替换w（a）：= {w_1（a），.，w_l（a）}给出了2l-正则图的随机塔的新模型：. Y_n--> Y_{n-1} -->. --> Y_0。对于F_m中的每个非循环子群D，（i）图$Y_n$具有有界的连通分支，（ii）这些连通分支构成扩张图族，（iii）D的闭包作为度量群Aut（T）的子群具有正的豪斯多夫维数.
英文摘要：
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).
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