Virtual retracts in groups acting on rooted trees

TL;DR

The paper investigates local retraction (LR) and related virtual retract concepts in groups acting on rooted trees, proving that finitely generated branch groups cannot have LR and linking LR to Thurston dynamics via iterated monodromy groups (IMG) of quadratic polynomials. It establishes a sharp LR characterization for quadratic post-critically finite polynomials: LR holds iff the Thurston orbifold is euclidean, which occurs exactly for the powering map and the Chebyshev polynomial. It further proves that periodic quadratics yield normal subgroups that are finitely generated and that the closures of their IMG groups have complete finitely generated Hausdorff spectrum, providing new pro-2 examples with rich spectra. The work intertwines branching group theory, kneading automata structure, and Thurston orbifold topology to reveal deep connections between profinite properties (LR, VRC) and dynamical systems, with precise classifications and finiteness results for IMG-derived groups.

Abstract

We study virtual retracts in groups acting on rooted trees. We show that finitely generated branch groups do not have the local retraction (LR) property. Furthermore, we specialize to iterated monodromy groups of post-critically finite quadratic complex polynomials and show that the (LR) property characterizes, among post-critically finite quadratic complex polynomials, those with a euclidean orbifold, i.e. the powering map and the Chebyshev polynomial. Lastly, we show that periodic quadratic complex polynomials provide new examples of pro-$2$ groups with complete finitely generated Hausdorff spectrum.

Theorems & Definitions (26)

• Theorem A
• Conjecture 2
• Theorem B
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 3.1
• Lemma 3.2
• proof