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Paul-Henry Leemann

Publications and source records attributed to Paul-Henry Leemann.

15 recordsLinked to original sources
arXiv2025

Non-closed subgroups of weakly branch groups

For a weakly branch group $G$ acting on a regular enough rooted tree, we provide two constructions of continuous families of distinct subgroups that are not closed in the profinite topology on $G$. On the one hand, we construct a continuous family of distinct non-closed subgroups such that each $H$ in the family is not ERF, that is, contains subgroups not closed in the profinite topology on $H$. On the other hand, under an additional assumption on $G$, we construct a continuous family of ERF subgroups which are not closed in the congruence (and in the profinite) topology on $G$.

math.GR↗
arXiv2025

Subgroup induction property for branch groups

The subgroup induction property is a property of self-similar groups acting on rooted trees introduced by Grigorchuk and Wilson in 2003 that appears to have strong implications on the structure of the groups possessing it. It was for example used in the proof that the first Grigorchuk group as well as the Gupta-Sidki 3-group are subgroup separable (locally extended residually finite) or to describe their finitely generated subgroups as well as their weakly maximal subgroups. However, until now, there were only two known examples of groups with this property, namely the first Grigorchuk group and the Gupta-Sidki 3-group. The aim of this article is twofold. First, we investigate various consequences of the subgroup induction property for branch groups, a particularly interesting class of self-similar groups. Notably, we show that finitely generated branch groups with the subgroup induction property must be torsion, just infinite and subgroup separable, and we establish conditions under which all their maximal subgroups are of finite index and all their weakly maximal subgroups are closed in the profinite topology. Then, we show that every torsion GGS group has the subgroup induction property, hence providing the first infinite family of examples of groups with this property.

math.GR↗
arXiv2024

Limits of Rauzy graphs of languages with subexponential complexity

To a subshift over a finite alphabet, one can naturally associate an infinite family of finite graphs, called its Rauzy graphs. We show that for a subshift of subexponential complexity the Rauzy graphs converge to the line $\mathbf{Z}$ in the sense of Benjamini-Schramm convergence if and only if its complexity function $p(n)$ is unbounded and satisfies $\lim_n\frac{p(n+1)}{p(n)} = 1$. We then apply this criterion to many examples of well-studied dynamical systems. If the subshift is moreover uniquely ergodic then we show that the limit of labelled Rauzy graphs if it exists can be identified with the unique invariant measure. In addition we consider an example of a non uniquely ergodic system recently studied by Cassaigne and Kaboré and identify a continuum of invariant measures with subsequential limits of labelled Rauzy graphs.

math.DS↗
arXiv2024

Weakly maximal subgroups of branch groups

Let $G$ be a branch group acting by automorphisms on a rooted tree $T$. Stabilizers of infinite rays in $T$ are examples of weakly maximal subgroups of $G$ (subgroups that are maximal among subgroups of infinite index), but in general they are not the only examples. In this note we describe two families of weakly maximal subgroups of branch groups. We show that, for the first Grigorchuk group as well as for the torsion GGS groups, every weakly maximal subgroup belongs to one of these families. The first family is a generalization of stabilizers of rays, while the second one consists of weakly maximal subgroups with a block structure. We obtain different equivalent characterizations of these families in terms of finite generation, the existence of a trivial rigid stabilizer, the number of orbit-closures for the action on the boundary of the tree or by the means of sections.

math.GR↗
arXiv2024

Wreath products of groups acting with bounded orbits

If $\textbf{S}$ is a subcategory of metric spaces, we say that a group G has property $B\textbf{S}$ if any isometric action on an $\textbf{S}$-space has bounded orbits. Examples of such subcategories include metric spaces, affine real Hilbert spaces, CAT(0) cube complexes, connected median graphs, trees or ultra-metric spaces. The corresponding properties $B\textbf{S}$ are respectively Bergman's property, property FH (which, for countable groups, is equivalent to the celebrated Kazhdan's property (T)), property FW (both for CAT(0) cube complexes and for connected median graphs), property FA and uncountable cofinality. Historically many of these properties were defined using the existence of fixed points. Our main result is that for many subcategories $\textbf{S}$, the wreath product $G\wr_XH$ has property $B\textbf{S}$ if and only if both $G$ and $H$ have property $B\textbf{S}$ and $X$ is finite. On one hand, this encompasses in a general setting previously known results for properties FH and FW. On the other hand, this also applies to the Bergman's property. Finally, we also obtain that $G\wr_XH$ has uncountable cofinality if and only if both $G$ and $H$ have uncountable cofinality and $H$ acts on $X$ with finitely many orbits.

math.GR↗
arXiv2022

Property FW and wreath products of groups: a simple approach using Schreier graphs

The group property FW stands in-between the celebrated Kazdhan's property (T) and Serre's property FA. Among many characterizations, it might be defined, for finitely generated groups, as having all Schreier graphs one-ended. It follows from the work of Y. Cornulier that a finitely generated wreath product $G\wr_XH$ has property~FW if and only if both $G$ and $H$ have property FW and $X$ is finite. The aim of this paper is to give an elementary, direct and explicit proof of this fact using Schreier graphs.

math.GR↗
arXiv2022

Finitely generated subgroups of branch groups and subdirect products of just infinite groups

The aim of this paper is to describe the structure of the finitely generated subgroups of a family of branch groups, which includes the first Grigorchuk group and the Gupta-Sidki 3-group. This description is made via the notion of block subgroup. We then use this to show that all groups in the above family are subgroup separable (LERF). These results are obtained as a corollary of a more general structural statement on subdirect products of just infinite groups.

math.GR↗
arXiv2021

Cayley graphs with few automorphisms: the case of infinite groups

We characterize the finitely generated groups that admit a Cayley graph whose only automorphisms are the translations, confirming a conjecture by Watkins from 1976. The proof relies on random walk techniques. As a consequence, every finitely generated group admits a Cayley graph with countable automorphism group. We also treat the case of directed graphs.

math.GR↗
arXiv2021

Most rigid representation and Cayley index of finitely generated groups

If $G$ is a group and $S$ a generating set, $G$ canonically embeds into the automorphism group of its Cayley graph and it is natural to try to minimize, over all generating sets, the index of this inclusion. This infimum is called the Cayley index of the group. In a recent series of works, we have characterized the infinite finitely generated groups with Cayley index $1$. We complement this characterization by showing that the Cayley index is $2$ in the remaining cases and is attained for a finite generating set.

math.GR↗
arXiv2019

Cayley graphs with few automorphisms

We show that every finitely generated group G with an element of order at least $(5rank(G))^{12}$ admits a locally finite directed Cayley graph with automorphism group equal to G. If moreover G is not generalized dihedral, then the above Cayley directed graph does not have bigons. On the other hand, if G is neither generalized dicyclic nor abelian and has an element of order at least $(2rank(G))^{36}$, then it admits an undirected Cayley graph with automorphism group equal to G. This extends classical results for finite groups and free products of groups. The above results are obtained as corollaries of a stronger form of rigidity which says that the rigidity of the graph can be observed in a ball of radius 1 around a vertex. This strong rigidity result also implies that the Cayley (di)graph covers very few (di)graphs. In particular, we obtain Cayley graphs of Tarski monsters which essentially do not cover other quasi-transitive graphs. We also show that a finitely generated group admits a locally finite labelled unoriented Cayley graph with automorphism group equal to itself if and only if it is neither generalized dicyclic nor abelian with an element of order greater than 2.

math.CO↗
arXiv2017

Lamplighter groups, de Bruijn graphs, spider-web graphs and their spectra

We describe the infinite family of spider-web graphs $S_{k,M,N }$, $k \geq 2$, $M \geq 1$ and $N \geq 0$, studied in physical literature as tensor products of well-known de Brujin graphs $B_{k,N}$ and cyclic graphs $C_M$ and show that these graphs are Schreier graphs of the lamplighter groups $L_k = Z/kZ \wr Z$. This allows us to compute their spectra and to identify the infinite limit of $S_{k,M,N}$, as $N, M \to\infty$, with the Cayley graph of the lamplighter group $L_k$. This is the final version of the article, taking in account comments from the referees and with an extended introduction.

math.CO↗
arXiv2016

Weakly maximal subgroups in regular branch groups

Let $G$ be a finitely generated regular branch group acting by automorphisms on a regular rooted tree $T$. It is well-known that stabilizers of infinite rays in $T$ (aka parabolic subgroups) are weakly maximal subgroups in $G$, that is, maximal among subgroups of infinite index. We show that, given a finite subgroup $Q\leq G$, $G$ possesses uncountably many automorphism equivalence classes of weakly maximal subgroups containing $Q$. In particular, for Grigorchuk-Gupta-Sidki type groups this implies that they have uncountably many automorphism equivalence classes of weakly maximal subgroups that are not parabolic.

math.GR↗
arXiv2015

Schreier graphs: transitivity and coverings

We give a characterization of isomorphisms between Schreier graphs in terms of the groups, subgroups and generating systems. This characterization may be thought as a graph analog of Mostow's rigidity theorem for hyperbolic manifolds. This allows us to give a transitivity criterion for Schreier graphs. Finally, we show that Tarski monsters satisfy a strong simplicity criterion.

math.CO↗