Virtual retracts in groups acting on rooted trees

TL;DR

This paper investigates virtual retracts in groups acting on rooted trees, revealing that certain branch groups lack the local retraction property and characterizing specific polynomial groups with this property.

Contribution

It demonstrates that finitely generated branch groups do not have the local retraction property and characterizes post-critically finite quadratic polynomials with this property.

Findings

Finitely generated branch groups lack the local retraction property.

The local retraction property characterizes powering maps and Chebyshev polynomials among quadratic polynomials.

Periodic quadratic polynomials yield new examples of pro-2 groups with complete finitely generated Hausdorff spectrum.

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.