On arithmetic properties of solvable Baumslag-Solitar groups

For $0<α\le 1$, we say that a sequence $(X_k)_{k>0}$ of $d$-regular graphs has property $D_α$ if there exists a constant $C>0$ such that $\mathrm{diam}(X_k)\ge C\cdot|X_k|^α$. We investigate property $D_α$ for arithmetic box spaces of the solvable Baumslag-Solitar groups $BS(1,m)$ (with $m\geq 2$): those are box spaces obtained by embedding $BS(1,m)$ into the upper triangular matrices in $GL_2(\mathbb{Z}[1/m])$ and intersecting with a family $M_{N_k}$ of congruence subgroups of $GL_2(\mathbb{Z}[1/m])$, where the levels $N_k$ are coprime with $m$ and $N_k|N_{k+1}$. We prove: - if an arithmetic box space has $D_α$, then $α\le\frac{1}{2}$~; - if the family $(N_k)_k$ of levels is supported on finitely many primes, the corresponding arithmetic box space has $D_{1/2}$~; - if the family $(N_k)_k$ of levels is supported on a family of primes with positive analytic primitive density, then the corresponding arithmetic box space does not have $D_α$, for every $α>0$. Moreover, we prove that if we embed $BS(1,m)$ in the group of invertible upper-triangular matrices $T_n(\mathbb{Z}[1/m])$, then every finite index subgroup of the embedding contains a congruence subgroup. This is a version of the congruence subgroup property (CSP).

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