arXiv subjects

Claudio Alexandre Piedade

Publications and source records attributed to Claudio Alexandre Piedade.

11 recordsLinked to original sources

arXiv2026

Coset geometries acting on their elements: The $j$-diagonals twisting

Let $\beta$ be a coset incidence system and fix a type $j$. We use the orbits of the group of $\beta$ on pairs of distinct $j$-elements to define a Coxeter graph. The action on the $j$-elements induces an action on the corresponding Coxeter group, so the twisting construction for coset incidence systems can be applied. The resulting coset geometry, called the $j$-diagonals twisting, is always a regular hypertope if $\beta$ is a regular hypertope. Using this, we show that finite regular hypertope whose diagram is a tree with all but one of the labels equal to four always exist. We also define an extension operation that combines this Coxeter graph with another given Coxeter graph. For regular polytopes, these constructions recover the twisting extensions of McMullen and Schulte.

math.CO↗

arXiv2026

The Geometry of Pattern Groups

Coset incidence geometries are an important tool which connect group theory and geometry. While the representations of the unitriangular group and its pattern subgroups have been studied extensively, the underlying geometric structures of these groups has remained largely unexplored. In this article, we construct the natural coset geometry associated with each pattern group over a finite field $\FF_q$, and prove geometrical properties of theses structures. We show how combinatorial features of their defining closed sets directly correspond to properties of their parabolic subgroups, in particular the intersection of parabolics and normality of the Borel subgroup. Finally, we characterize the automorphisms of these geometries and examine how pattern geometries behave under change of field within the same characteristic $p$.

math.GR↗

arXiv2026

The geometry of wreath and semi-direct products

Coset geometries are incidence geometries constructed from a group $G$ and a system of subgroups $(G_i)_{i \in I}$ of subgroups of $G$. For any algebraic group operation, it is then natural to wonder whether it can be extended to the framework of coset geometries. This has been achieved in the case of the halving (\cite{halving}) and in the case of free (amalgamated) products, HNN-extensions, and semi-direct products (\cite{piedade2025group}). In this article, we explore more deeply two operations related to semi-direct products: the twisting and the wreath product. We show that these operations extend to coset geometries in such a way that they preserve key properties, such as flag-transitivity, residual-connectedness and being thin. In particular, we can apply twistings and wreath products to polytopes and hypertopes. Doing so, we show that there exists regular polytopes and hypertopes for almost-simple group with socle a sporadic simple group.

math.GR↗

arXiv2025

From Group Operations to Geometric Structures: Amalgamations, HNN-Extensions, and Twisting in Coset Geometries

Coset incidence geometries, introduced by Jacques Tits, provide a versatile framework for studying the interplay between group theory and geometry. In this article, we build upon that idea by extending classical group-theoretic constructions (amalgamated products, HNN-extensions, semi-direct products, and twisting) to the setting of coset geometries. This gives a general way to glue together incidence geometries in various ways. This provides a general framework for combining or gluing incidence geometries in different ways while preserving essential properties such as flag-transitivity and residual connectedness. Using these techniques, we analyze families of Shephard groups, which generalize both Coxeter and Artin-Tits groups, and their associated simplicial complexes. Our results also point to the existence of a Bass-Serre theory for coset geometries and of a fundamental geometry of a graph of coset geometries.

math.GR↗

arXiv2024

Regular polytopes of rank $n/2$ for transitive groups of degree $n$

Previous research established that the maximal rank of the abstract regular polytopes whose automorphism group is a transitive proper subgroup of $\mbox{S}_n$ is $n/2 + 1$. Up to isomorphism and duality, when $n\geq 12$, there are only two polytopes attaining this rank and they occur when $n/2$ is odd, and hence have even rank. In this paper, we investigate the case where the rank is equal to $n/2$ ($n\geq 14$). Our analysis suggests that reducing the rank by one results in a substantial increase in the number of regular polytopes.

math.CO↗

arXiv2024

Constructing new geometries: a generalized approach to halving for hypertopes

Given a residually connected incidence geometry $Γ$ that satisfies two conditions, denoted $(B_1)$ and $(B_2)$, we construct a new geometry $H(Γ)$ with properties similar to those of $Γ$. This new geometry $H(Γ)$ is inspired by a construction of Percsy, Percsy and Leemans [1]. We show how $H(Γ)$ relates to the classical halving operation on polytopes, allowing us to generalize the halving operation to a broader class of geometries, that we call non-degenerate leaf hypertopes. Finally, we apply this generalization to cubic toroids in order to generate new examples of regular hypertopes.

math.CO↗

arXiv2023

String C-group representations of transitive Groups: a case study with degree $11$

In this paper we give a non-computer-assisted proof of the following result: if $G$ is an even transitive group of degree $11$ and has a string C-group representation with rank $r\in\{4,5\}$ then $G\cong\PSL_2(11)$. Moreover this string C-group is the group of automorphisms of the rank $4$ polytope known as the $11$-cell. The insights gained from this case study include techniques and observations concerning permutation representation graphs of string C-groups. The foundational lemmas yield a natural and intuitive understanding of these groups. These and similar approaches can be replicated and are applicable to the study of other transitive groups.

math.GR↗

arXiv2023

The degrees of the orientation-preserving automorphism groups of toroidal maps and hypermaps

This paper is an exploration of the faithful transitive permutation representations of the orientation-preserving automorphisms groups of highly symmetric toroidal maps and hypermaps. The main theorems of this paper give a list of all possible degrees of these specific groups. This extends prior accomplishments of the authors, wherein their focus was confined to the study of the automorphisms groups of toroidal regular maps and hypermaps. In addition the authors bring out the recently developed {\sc GAP} package {\sc corefreesub} that can be used to find faithful transitive permutation representations of any group. With the aid of this powerful tool, the authors show how Schreier coset graphs of the automorphism groups of toroidal maps and hypermaps can be easily constructed.

math.GR↗

arXiv2021

Infinite families of hypertopes from centrally symmetric polytopes

We construct infinite families of abstract regular polytopes of type $\{4,p_1,\ldots,p_{n-1}\}$ from extensions of centrally symmetric spherical abstract regular $n$-polytopes. In addition, by applying the halving operation, we obtain infinite families of both locally spherical and locally toroidal regular hypertopes of type $\left\{{p_1 \atop p_1},\ldots,p_{n-1}\right\}$.

math.CO↗

arXiv2020

The Degrees Of Toroidal Regular Proper Hypermaps

Recently the classification of all possible faithful transitive permutation representations of the group of symmetries of a regular toroidal map was accomplished. In this paper we complete this investigation on a surface of genus 1 considering the group of a regular toroidal hypermap of type $(3,3,3)$ that is a subgroup of index $2$ of the group of symmetries of a toroidal map of type $\{6,3\}$.

math.GR↗

arXiv2019

Faithful permutation representations of toroidal regular maps

In this paper we list all possible degrees of a faithful transitive permutation representation of the group of symmetries of a regular map of types $\{4,4\}$ and $\{3,6\}$ and we give examples of graphs, called CPR-graphs, representing some of these permutation representations.

math.AG↗