Random walk on group extensions
Alireza Salehi Golsefidy, Srivatsa Srinivas · Transactions of the American Mathematical Society · 2024
We study random walks on various group extensions. Under certain bounded generation and bounded scaled conditions, we estimate the spectral gap of a random walk on a quasi-random-by-nilpotent group in terms of the spectral gap of its projection to the quasi-random part. We also estimate the spectral gap of a random-walk on a product of two quasi-random groups in terms of the spectral gap of its projections to the given factors. Based on these results, we estimate the spectral gap of a random walk on the F q {\mathbb {F}}_q -points of a perfect algebraic group G {\mathbb {G}} in terms of the spectral gap of its projections to the almost simple factors of the semisimple quotient of G {\mathbb {G}} . These results extend a work of Lindenstrauss and Varjú and an earlier work of the authors. Moreover, using a result of Breuillard and Gamburd, we show that there is an infinite set P \mathcal {P} of primes of density one such that, if k k is a positive integer and G = U ⋊ ( SL 2 ) Q m {\mathbb {G}}={\mathbb {U}}\rtimes (\operatorname {SL}_2)_{\mathbb {Q}}^m is a perfect group and U {\mathbb {U}} is a unipotent group, then the family of all the Cayley graphs of G ( Z / ∏ i = 1 k p i Z ) {\mathbb {G}}({\mathbb {Z}}/\prod _{i=1}^{k}p_i{\mathbb {