Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On arithmetic properties of solvable Baumslag-Solitar groups   For $0<\alpha\le 1$, we say that a sequence $(X_k)_{k>0}$ of $d$-regular
graphs has property $D_\alpha$ if there exists a constant $C>0$ such that
$\mathrm{diam}(X_k)\ge C\cdot|X_k|^\alpha$. We investigate property $D_\alpha$
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 $D_\alpha$, then $\alpha\le\frac{1}{2}$~;
  - 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 $D_\alpha$, for every $\alpha>0$.
  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).

33
The categories of this paper are: plasm-ph
16
The categories of this paper are: stat
23
The categories of this paper are: solv-int
4
The categories of this paper are: cond-mat
10
The categories of this paper are: q-bio
0
The categories of this paper are: hep-ph
9
The categories of this paper are: nlin
6
The categories of this paper are: astro-ph
22
The categories of this paper are: chao-dyn
25
The categories of this paper are: patt-sol
21
The categories of this paper are: adap-org
30
The categories of this paper are: q-alg
7
The categories of this paper are: hep-th
24
The categories of this paper are: dg-ga
14
The categories of this paper are: math-ph
34
The categories of this paper are: chem-ph