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Ilaria Castellano

Publications and source records attributed to Ilaria Castellano.

At least 19 recordsLinked to original sources
arXiv2026

Generalisable presentations and compactness properties of locally compact right-angled Artin groups

We propose the systematic study of presentations that can be generalised over a continuous open group monomorphism. Presentations with this property can turn well-known presentations such as those for as orientable surface groups, Artin groups, and some Thompson groups, into topological groups with a prescribed open subgroup. Later we focus on right-angled Artin groups (RAAGs) and introduce a notion of topological RAAGs. Our approach differs from lattice envelopes and produces examples of locally compact (LC) groups that contain RAAGs as discrete subgroups, but generally not as lattices. We investigate some geometric aspects of topological RAAGs, with a special emphasis on compactness properties of LC ones. This includes a study of universal Salvetti-type complexes which may be of independent interest. These complexes share some properties with buildings. Although in some cases they are CAT(0) cube complexes and provide models for classifying spaces, in other cases they are not even uniquely geodesic. For a large class of examples we establish high connectivity properties for these complexes. This yields novel examples of LC groups with prescribed compactness properties or rational cohomological dimension. We note that the Bestvina-Brady machinery does not automatically generalise to this setting; nevertheless, we extend the Bieri-Stallings construction to obtain totally disconnected locally compact (TDLC) groups of type $FP_n$ but not $FP_{n+1}$. Along the way we record counterparts of cohomological results, such as a Mayer-Vietoris sequence and Künneth formula in discrete (co)homology for TDLC groups, which have not appeared elsewhere in the literature. Despite our non-discrete LC focus we obtain, as by-product, new examples of discrete groups with controlled finiteness properties including, for every $n \geq 1$, a Thompson-like Bieri-Stallings group of type $F_n$ but not $F_{n+1}$.

math.GR↗
arXiv2026

On cohomological dimensions of totally disconnected locally compact groups

In this paper, we introduce Mackey functors for a t.d.l.c. group and define the cohomological dimension of this group over the Mackey category. We then compare this dimension to the rational discrete cohomological dimension defined by Castellano and Weigel, as well as to the Bredon cohomological dimension of that t.d.l.c. group with respect to the family of compact open subgroups. We also extend results about the geometric dimension of a t.d.l.c. group.

math.GR↗
arXiv2025

Geometric invariants of TDLC completions

Recently, Bonn and Sauer showed that, from the point of view of compactness properties, the Schlichting completion of a Hecke pair $(Γ,Λ)$ behaves precisely as if it were the quotient of $Γ$ by $Λ$. Motivated by this result, we prove that a similar phenomenon holds for the $Σ$-sets. More generally, we relate the $Σ$-sets of every TDLC completion of a Hecke pair $(Γ,Λ)$ to the $Σ$-sets of $Γ$ whenever $Λ$ satisfies suitable compactness properties. We provide applications to TDLC completions of Baumslag-Solitar groups and certain groups of upper triangular matrices studied by Schesler.

math.GR↗
arXiv2025

Some invariants of totally disconnected locally compact groups: cohomology and combinatorics

The paper investigates two invariants for totally disconnected locally compact groups: the number of ends and the rational discrete cohomological dimension. For such a compactly generated group $G$ it is shown that its number of ends can be expressed in terms of the rational discrete cohomology of $G$. If $G$ is suitably acting on a building the number of ends and the rational cohomological dimension of $G$ are related to those of the Weyl group associated to the building. In special cases, we are also able to compare the rational discrete cohomological dimension of $G$ to the flat-rank of $G$. Moreover, examples of groups for which these two invariants coincide are given. Our approach leverages the combinatorics of Coxeter groups, yielding new results of independent interest in Coxeter theory. Finally, in the class of totally disconnected locally compact groups acting properly and cocompactly on locally finite buildings, an accessibility result is proved: we explicitly construct a cocompact proper action on a tree if the rational discrete cohomological dimension is one.

math.GR↗
arXiv2025

Unimodular totally disconnected locally compact groups of rational discrete cohomological dimension one

It is shown that a Stallings--Swan theorem holds in a totally disconnected locally compact (= t.d.l.c.) context (cf. Thm. B). More precisely, a compactly generated $\mathcal{CO}$-bounded t.d.l.c. group $G$ of rational discrete cohomological dimension less than or equal to $1$ must be isomorphic to the fundamental group of a finite graph of profinite groups. This result generalises Dunwoody's rational version of the classical Stallings--Swan theorem to t.d.l.c. groups. The proof of Theorem B is based on the fact that a compactly generated unimodular t.d.l.c. group with rational discrete cohomological dimension $1$ has necessarily non-positive Euler--Poincaré characteristic (cf. Thm. H).

math.GR↗
arXiv2024

The Hattori-Stallings rank, the Euler-Poincaré characteristic and zeta functions of totally disconnected locally compact groups

For a unimodular totally disconnected locally compact group $G$ we introduce and study an analogue of the Hattori-Stallings rank $\tildeρ(P)\in\mathbf{h}_G$ for a finitely generated projective rational discrete left $\mathbb Q[G]$-module $P$. Here $\mathbf{h}_G$ denotes the $\mathbb Q$-vector space of left invariant Haar measures of $G$. Indeed, an analogue of Kaplansky's theorem holds in this context (cf. Theorem A). As in the discrete case, using this rank function it is possible to define a rational discrete Euler-Poincaré characteristic $\tildeχ_G$ whenever $G$ is a unimodular totally disconnected locally compact group of type $\mathrm{FP}_\infty$ of finite rational discrete cohomological dimension. E.g., when $G$ is a discrete group of type $\mathrm{FP}$, then $\tildeχ_G$ coincides with the ''classical'' Euler-Poincaré characteristic times the counting measure $μ_{\{1\}}$. For a profinite group $\mathcal{O}$, $\tildeχ_{\mathcal{O}}$ equals the probability Haar measure $μ_{\mathcal{O}}$ on $\mathcal{O}$. Many more examples are calculated explicitly (cf. Example 1.7 and Section 5). In the last section, for a totally disconnected locally compact group $G$ satisfying an additional finiteness condition, we introduce and study a formal Dirichlet series $ζ_{_{G,\mathcal{O}}}(s)$ for any compact open subgroup $\mathcal{O}$. In several cases it happens that $ζ_{_{G,\mathcal{O}}}(s)$ defines a meromorphic function $\tildeζ_{_{G,\mathcal{O}}}\colon \mathbb{C} \to\bar{\mathbb C}$ of the complex plane satisfying miraculously the identity $\tildeχ_G=\tildeζ_{_{G,\mathcal{O}}}(-1)^{-1}\cdotμ_{\mathcal{O}}$. Here $μ_{\mathcal{O}}$ denotes the Haar measure of $G$ satisfying $μ_{\mathcal{O}}(\mathcal{O})=1$.

math.GR↗
arXiv2022

Weakly weighted generalised quasi-metric spaces and semilattices

Motivated by recent applications to entropy theory in dynamical systems, we generalise notions introduced by Matthews and define weakly weighted and componentwisely weakly weighted (generalised) quasi-metrics. We then systematise and extend to full generality the correspondences between these objects and other structures arising in theoretical computer science and dynamics. In particular, we study the correspondences with weak partial metrics, and, if the underlying space is a semilattice, with invariant (generalised) quasi-metrics satisfying the descending path condition, and with strictly monotone semi(-co-)valuations. We conclude discussing, for endomorphisms of generalised quasi-metric semilattices, a generalisation of both the known intrinsic semilattice entropy and the semigroup entropy.

cs.IT↗
arXiv2021

Subgroups, hyperbolicity and cohomological dimension for totally disconnected locally compact groups

This article is part of the program of studying large-scale geometric properties of totally disconnected locally compact groups, TDLC-groups, by analogy with the theory for discrete groups. We provide a characterization of hyperbolic TDLC-groups, in terms of homological isoperimetric inequalities. This characterization is used to prove the main result of the article: for hyperbolic TDLC-groups with rational discrete cohomological dimension $\leq 2$, hyperbolicity is inherited by compactly presented closed subgroups. As a consequence, every compactly presented closed subgroup of the automorphism group $\mathrm{Aut}(X)$ of a negatively curved locally finite $2$-dimensional building $X$ is a hyperbolic TDLC-group, whenever $\mathrm{Aut}(X)$ acts with finitely many orbits on $X$. Examples where this result applies include hyperbolic Bourdon's buildings. We revisit the construction of small cancellation quotients of amalgamated free products, and verify that it provides examples of hyperbolic TDLC-groups of rational discrete cohomological dimension $2$ when applied to amalgamated products of profinite groups over open subgroups. We raise the question of whether our main result can be extended to locally compact hyperbolic groups if rational discrete cohomological dimension is replaced by asymptotic dimension. We prove that this is the case for discrete groups and sketch an argument for TDLC-groups.

math.GR↗
arXiv2020

Subgroups of pro-$p$ $\mathrm{PD}^3$-groups

We study 3-dimensional Poincaré duality pro-$p$ groups in the spirit of the work by Robert Bieri and Jonathan Hillmann, and show that if such a pro-$p$ group $G$ has a nontrivial finitely presented subnormal subgroup of infinite index, then either the subgroup is cyclic and normal, or the subgroup is cyclic and the group is polycyclic, or the subgroup is Demushkin and normal in an open subgroup of $G$. Also, we describe the centralizers of finitely generated subgroups of 3-dimensional Poincaré duality pro-$p$ groups.

math.GR↗
arXiv2020

Intrinsic entropy for generalized quasimetric semilattices

We introduce the notion of intrinsic semilattice entropy $\widetilde h$ in the category $\mathcal L_{qm}$ of generalized quasimetric semilattices and contractive homomorphisms. By using appropriate categories $\mathfrak X$ and functors $F:\mathfrak X\to\mathcal L_{qm}$ we find specific known entropies $\widetilde h_\mathfrak X$ on $\mathfrak X$ as intrinsic functorial entropies, that is, as $\widetilde h_\mathfrak X=\widetilde h\circ F$. These entropies are the intrinsic algebraic entropy, the algebraic and the topological entropies for locally linearly compact vector spaces, the topological entropy for locally compact totally disconnected groups and the algebraic entropy for locally compact compactly covered abelian groups.

math.GR↗
arXiv2019

A property of the lamplighter group

We show that the inert subgroups of the lamplighter group fall into exactly five commensurability classes. The result is then connected with the theory of totally disconnected locally compact groups and with algebraic entropy.

math.GR↗
arXiv2019

Finiteness properties of totally disconnected locally compact groups

In this paper we investigate finiteness properties of totally disconnected locally compact groups for general commutative rings $R$, in particular for $R = \mathbb{Z}$ and $R= \mathbb{Q}$. We show these properties satisfy many analogous results to the case of discrete groups, and we provide analogues of the famous Bieri's and Brown's criteria for finiteness properties and deduce that both $FP_n$-properties and $F_n$-properties are quasi-isometric invariant. Moreover, we introduce graph-wreath products in the category of totally disconnected locally compact groups and discuss their finiteness properties.

math.GR↗
arXiv2018

The corank of a flow over the category of linearly compact vector spaces

For a topological flow $(V,ϕ)$ - i.e., $V$ is a linearly compact vector space and $ϕ$ a continuous endomorphism of $V$ - we gain a deep understanding of the relationship between $(V,ϕ)$ and the Bernoulli shift: a topological flow $(V,ϕ)$ is essentially a product of one-dimensional left Bernoulli shifts as many as $\mathrm{ent}^*(V,ϕ)$ counts. This novel comprehension brings us to introduce a notion of corank for topological flows designed for coinciding with the value of the topological entropy of $(V,ϕ)$. As an application, we provide an alternative proof of the so-called Bridge Theorem for locally linearly compact vector spaces connecting the topological entropy to the algebraic entropy by means of Lefschetz duality.

math.GR↗
arXiv2017

Rational discrete cohomology for totally disconnected locally compact groups

Rational discrete cohomology and homology for a totally disconnected locally compact group $G$ is introduced and studied. The $\mathrm{Hom}$-$\otimes$ identities associated to the rational discrete bimodule $\mathrm{Bi}(G)$ allow to introduce the notion of rational duality groups in analogy to the discrete case. It is shown that semi-simple groups defined over a non-discrete, non-archimedean local field are rational t.d.l.c. duality groups, and the same is true for certain topological Kac-Moody groups. However, Y. Neretin's group of spheromorphisms of a locally finite regular tree is not even of finite rational discrete cohomological dimension. For a unimodular t.d.l.c. group $G$ of type $\mathrm{FP}$ it is possible to define an Euler-Poincaré characteristic $χ(G)$ which is a rational multiple of a Haar measure. This value is calculated explicitly for Chevalley groups defined over a non-discrete, non-archimedean local field $K$ and some other examples.

math.GR↗
arXiv2017

Rational discrete first degree cohomology for totally disconnected locally compact groups

It is well-known that the existence of more than two ends in the sense of J.R. Stallings for a finitely generated discrete group $G$ can be detected on the cohomology group $\mathrm{H}^1(G,R[G])$, where $R$ is either a finite field, the ring of integers or the field of rational numbers. It will be shown (cf. Theorem A*) that for a compactly generated totally disconnected locally compact group $G$ the same information about the number of ends of $G$ in the sense of H. Abels can be provided by $\mathrm{dH}^1(G,\mathrm{Bi}(G))$, where $\mathrm{Bi}(G)$ is the rational discrete standard bimodule of $G$, and $\mathrm{dH}^\bullet(G,\_)$ denotes rational discrete cohomology as introduced in [6]. As a consequence one has that the class of fundamental groups of a finite graph of profinite groups coincides with the class of compactly presented totally disconnected locally compact groups of rational discrete cohomological dimension at most 1 (cf. Theorem B).

math.GR↗
arXiv2017

Topological entropy for locally linearly compact vector spaces

In analogy to the topological entropy for continuous endomorphisms of totally disconnected locally compact groups, we introduce a notion of topological entropy for continuous endomorphisms of locally linearly compact vector spaces. We study the fundamental properties of this entropy and we prove the Addition Theorem, showing that the topological entropy is additive with respect to short exact sequences. By means of Lefschetz Duality, we connect the topological entropy to the algebraic entropy in a Bridge Theorem.

math.GR↗