On finitely generated Hausdorff spectra of groups acting on rooted trees

Abstract

We answer two longstanding questions of Klopsch (1999) and Shalev (2000) by proving that the finitely generated Hausdorff spectrum of the closure of a finitely generated regular branch group with respect to the level-stabilizer filtration is the full interval [0,1]. Previous work related to the above questions include the celebrated result of Ab\'ert and Vir\'ag on the completeness of the finitely generated Hausdorff spectrum of the group of $p$-adic automorphisms $W_p$. We extend their result to any level-transitive iterated wreath product acting on a regular rooted tree, and more importantly, provide the first explicit examples of finitely generated subgroups with prescribed Hausdorff dimension. In fact, in $W_p$ such subgroups can be taken to be 2-generated, giving further evidence to a conjecture of Ab\'ert and Vir\'ag. Furthermore, we extend a well-known result of Klopsch (1999) for branch groups to closed level-transitive groups with fully dimensional rigid stabilizers: these have full Hausdorff spectrum with respect to the level-stabilizer filtration. In turn, we determine, for the first time, the Hausdorff spectum of many well-studied families of weakly branch groups. As an additional new contribution, we define {\it nice filtrations} (which include the level-stabilizer filtration). In fact, the first and the last result can be proven more generally with respect to these filtrations (under possibly some further assumptions). Finally, we also consider two standard filtrations, namely the $2$-central lower series, and the dimension subgroup or Jenning-Zassenhaus series in the closure of the first Grigorchuck group, and show that its finitely generated Hausdorff spectra with respect to these filtrations is the full interval [0,1], answering a recent question of de las Heras and Thillaisundaram (2022). This is obtained via a reduction to properties concerning to nice filtrations.