On arithmetic properties of solvable Baumslag-Solitar groups (original) (raw)
For 0 < α ≤ 1, we say that a sequence (Xk )k>0 of d-regular graphs has property Dα if there exists a constant C > 0 such that diam(Xk ) ≥ C · |Xk | . We investigate property Dα for arithmetic box spaces of the solvable Baumslag-Solitar groups BS (1,m) (withm ≥ 2): those are box spaces obtained by embedding BS (1,m) into the upper triangular matrices in GL2 (Z[1/m]) and intersecting with a familyMNk of congruence subgroups of GL2 (Z[1/m]), where the levels Nk are coprime withm and Nk |Nk+1. We prove: • if an arithmetic box space has Dα , then α ≤ 2 ; • if the family (Nk )k of levels is supported on finitely many primes, the corresponding arithmetic box space has D1/2 ; • if the family (Nk )k of levels is supported on a family of primes with positive analytic primitive density, then the corresponding arithmetic box space does not have Dα , for every α > 0. Moreover, we prove that if we embed BS (1,m) in the group of invertible upper-triangular matricesTn (Z[1/m]), then eve...
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