arXiv subjects

Jon Merladet Urigüen

Publications and source records attributed to Jon Merladet Urigüen.

3 recordsLinked to original sources

arXiv2026

Virtual retracts in groups acting on rooted trees

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.

math.GR↗

arXiv2025

Property (VRC) and virtual fibering for amalgamated free products

This paper focuses on studying properties of amalgamated free products $G=G_1*_{G_0} G_2$, where the amalgamated subgroup $G_0$ is virtually cyclic. First, we prove that if the factors $G_1$ and $G_2$ are finitely generated virtually abelian groups then $G$ can be mapped to another virtually abelian group so that this homomorphism is injective on each factor. We then present several applications of this result. In particular, we show that if $G_1$ and $G_2$ have property (VRC) (that is, every cyclic subgroup is a virtual retract), then the same is true for $G$. We also prove that $G$ inherits some residual properties (such as residual finiteness or virtual residual solvability) from the factors $G_i$, provided $G_0$ is a virtual retract of $G_i$, for $i=1,2$. Finally, we give necessary and sufficient conditions for $G$ to be (virtually) $F_m$-fibered. In particular, we fully characterize when an amalgamated product of two (finitely generated free)-by-cyclic groups over a cyclic subgroup is free-by-cyclic or virtually free-by-cyclic.

math.GR↗

arXiv2025

Virtual retractions in free constructions

A group $G$ has property (VRC) if every cyclic subgroup is a virtual retract. This property is stable under many standard group-theoretic constructions and is enjoyed by all virtually special groups (in the sense of Haglund and Wise). In this paper we study property (VRC) for fundamental groups of finite graphs of groups. Our main criterion shows that the fundamental group of a finite graph of finitely generated virtually abelian groups has (VRC) if and only if it has a homomorphism to a Euclidean-by-finite group that is injective on all vertex groups. This result allows us to determine property (VRC) for such groups using basic tools from Euclidean Geometry and Linear Algebra. We use it to produce examples and to give sufficient criteria for fundamental groups of finite graphs of finitely generated abelian groups with cyclic edge groups to have (VRC). In the last two sections and in the appendix we give applications of property (VRC). We show that if a fundamental group of a finite graph of groups with finitely generated virtually abelian vertex groups has (VRC) then it is CAT($0$). We also show that tubular groups with (VRC) are virtually free-by-cyclic and virtually special.

math.GR↗