On arithmetic properties of solvable Baumslag–Solitar groups

Abstract

Abstract For 0 < α ≤ 1 0<\alpha\leq 1 , we say that a sequence ( X k ) k > 0 (X_{k})_{k>0} of 𝑑-regular connected graphs has property D α D_{\alpha} if there exists a constant C > 0 C>0 such that diam ⁡ ( X k ) ≥ C ⋅ | X k | α \operatorname{diam}(X_{k})\geq C\cdot\lvert X_{k}\rvert^{\alpha} . We investigate property D α D_{\alpha} for arithmetic box spaces of the solvable Baumslag–Solitar groups BS ⁡ ( 1 , m ) \operatorname{BS}(1,m) (with m ≥ 2 m\geq 2 ): these are box spaces obtained by embedding BS ⁡ ( 1 , m ) \operatorname{BS}(1,m) into the upper triangular matrices in GL 2 ⁢ ( Z ⁢ [ 1 / m ] ) \mathrm{GL}_{2}(\mathbb{Z}[1/m]) and intersecting with a family M N k M_{N_{k}} of congruence subgroups of GL 2 ⁢ ( Z ⁢ [ 1 / m ] ) \mathrm{GL}_{2}(\mathbb{Z}[1/m]) , where the levels N k N_{k} are coprime with 𝑚 and N k ∣ N k + 1 N_{k}\mid N_{k+1} . We prove that if an arithmetic box space has D α D_{\alpha} , then α ≤ 1 2 \alpha\leq\frac{1}{2} ; if the family ( N k ) k (N_{k})_{k} of levels is supported on finitely many primes, the corresponding arithmetic box space has D 1 / 2 D_{\smash{1/2}} ; if the family ( N k ) k (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 D_{\alpha} for every α > 0 \alpha>0 . Moreover, we prove that if we embed BS ⁡ ( 1 , m ) \operatorname{BS}(1,m) in the group of invertible upper-triangular matrices T n ⁢ ( Z ⁢ [ 1 / m ] ) 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).