Locally Random Groups
Group Theory
2020-09-02 v1 Combinatorics
Dynamical Systems
Abstract
In this work, we will introduce and study the notion of local randomness for compact metric groups. We prove a mixing inequality as well as a product result for locally random groups under an additional dimension condition on the volume of small balls and provide several examples of such groups. In particular, this leads to new examples of groups satisfying such a mixing inequality. In the same context, we will develop a Littlewood-Paley decomposition and explore its connection to the existence of the spectral gap for random walks. Moreover, under the dimension condition alone, we will prove a multi-scale entropy gain result `a la Bourgain-Gamburd and Tao.
Cite
@article{arxiv.2008.13645,
title = {Locally Random Groups},
author = {Keivan Mallahi-Karai and Amir Mohammadi and Alireza Salehi Golsefidy},
journal= {arXiv preprint arXiv:2008.13645},
year = {2020}
}
Comments
41 pages