arXiv subjects

Miguel Donoso-Echenique

Publications and source records attributed to Miguel Donoso-Echenique.

5 recordsLinked to original sources

arXiv2026

Free Burnside groups of large odd exponent have cost 1

We prove that for every non-elementary torsion-free hyperbolic group $H$ and sufficiently large odd $n>1$, the quotient $H/H^n$ has cost $1$. In particular, when $H$ is the free group of rank $m\ge 2$, we obtain that the free Burnside group $B(m,n)$ has cost $1$ for every odd exponent $n\geq 665$. It follows that $B(m,n)$ is anti-treeable, and that its first $\ell^2$-Betti number $β_1^{(2)}(B(m,n))$ vanishes. The latter recovers and extends a result of Feldkamp and Kionke [Proc. Amer. Math. Soc., 2023], who proved that $β_1^{(2)}(B(m,p))=0$ for all sufficiently large prime exponents $p$.

math.GR↗

arXiv2026

Random operators, spectral measures, and local empirical convergence in sofic groups

In this paper, we consider the problem of approximating the spectral distribution for a class of random operators over sofic groups. For this purpose, we make use of the concept of locally and empirically converging measures defined by Austin. We establish weak convergence of the density of states measures along random finite-volume analogs. For operators taking finitely many rational values, we prove a Lück type approximation theorem yielding pointwise convergence of the spectral measures. In the wider context of arbitrary complex coefficients, we show pointwise convergence of the spectral distribution functions along adapted approximants with varying rational coefficients. Our results apply to the class of periodically approximable groups as defined by Bowen. More generally, we show that every invariant probability measure on a finite-state configuration space that arises as a weak-$\ast$ limit of periodic measures admits an approximation in the local and empirical sense.

math.SP↗

arXiv2025

Expansivity of algebraic semigroup actions

For a semigroup $S$ and a right $\mathbb{Z}[S]$-submodule $J\leq \mathbb{Z}[S]^n$, we study expansivity of the algebraic action of $S$ induced on the Pontryagin dual of $\mathbb{Z}[S]^n/J$. We completely determine the class of semigroups for which expansivity of this action is characterized by the triviality of $J^\perp$, in terms of the existence of a finite subset $K\subseteq S$ such that $S=KS$. This condition is satisfied in particular by every monoid, and more generally by every semigroup with a left unital convolution Banach algebra, in which case we are able to extend the characterization of expansivity in terms of an invertibility condition when $J$ is finitely generated. We also exhibit examples of non-unital semigroups satisfying the hypotheses of our results.

math.DS↗

arXiv2025

Natural extensions of embeddable semigroup actions

Semigroup actions and their invertible extensions are discussed. First, we develop a theory of natural extensions for continuous actions of countable, embeddable semigroups. Second, we demonstrate that not every surjective such action of a semigroup, which embeds into a group and generates it, can be extended to an action of said group, and that this phenomenon is specific to non-reversible semigroups. Furthermore, we characterize the free group on a semigroup (the group together with the embedding) as the unique pair that always admits such an extension, showing that both the choice of the receiving group and the embedding are crucial for this construction. Next, we prove that the classical notion of a natural extension -- requiring all other invertible extensions to factor through it -- only works in the context of compact extensions of left reversible semigroup actions and fails outside of it, thus providing a characterization of left reversibility. We finish by briefly studying topological dynamical properties of the natural extension in the amenable case.

math.DS↗

arXiv2025

Extensibility and denseness of periodic semigroup actions

We study periodic points and finitely supported invariant measures for continuous semigroup actions. Introducing suitable notions of periodicity in both topological and measure-theoretical contexts, we analyze the space of invariant Borel probability measures associated with these actions. For embeddable semigroups, we establish a direct relationship between the extensibility of invariant measures to the free group on the semigroup and the denseness of finitely supported invariant measures. Applying this framework to shift actions on the full shift, we prove that finitely supported invariant measures are dense for every left amenable semigroup that is residually a finite group and for every finite-rank free semigroup.

math.DS↗