Teoria Geometrica dei Gruppi Spazi CAT(0), Teorema di Gromov e oriented right-angled Artin groups

TL;DR

This thesis explores spaces with bounded above curvature, focusing on CAT(0) spaces, Gromov's theorem for cubical complexes, and introduces a new class of discrete groups extending right-angled Artin groups.

Contribution

It presents new applications of CAT(0) geometry in combinatorial algebra and introduces a novel class of discrete groups with associated cubical CAT(0)-complexes.

Findings

Application of CAT(0) spaces in algebraic contexts

Extension of Gromov's theorem to new cubical complexes

Introduction of a new class of discrete groups

Abstract

The aim of this thesis is to present the notion of spaces whose curvature is bounded above, and to give some of its application in the context of Combinatorial Algebra. The thesis is made of two parts, one of theoretic purpose, and the other applicative. In particular, we present an application to the notion of $CAT(0)$-space and one to the Gromov Theorem for cubical complexes. In the final part we present a new class of discrete groups generalizing the well-known right-angled Artin groups. We concoct a cubical CAT(0)-complex on which the group acts, from whom we extract some algebraic properties of such groups.