Dans la première partie, on étudie la probabilité de retour des groupes métabéliens de type fini. On donne une caractérisation des tels groupes avec grande probabilité de retour en des termes purement algébriques, à l’aide de la dimension de Krull. Cela nécessite, pour les groupes métabéliens, une variation d’un théorème de Kaloujnine et Krasner qui respecte cette dimension. Au passage, on obtient des bornes inférieures et supérieures sur la probabilité de retour des groupes métabéliens en fonction de la dimension de Krull. La seconde partie concerne les profils isopérimétriques des groupes localement compacts compactement engendrés, qu’on utilise pour caractériser l’existence d’une suite de paires de Følner. On démontre que le profil isopérimétrique augmente lorsqu’on passe au quotient, avec des constantes indépendantes de l’échelle, améliorant une théorème de Tessera. Combinant les deux, on obtient que l’existence de suites de paires de Følner passe au quotient. On montre qu’elle passe au sous-groupe fermé, généralisant un résultat correspondant d’Erschler pour les groupes de type fini. Cela permet d’obtenir une preuve plus auto-contenue du théorème principal de la première partie.La troisième partie est un travail en commun avec Kropholler dans lequel on étudie la structure des groupes résolubles de rang sans torsion infini n’ayant pas de section isomorphe à ZwrZ. On en déduit qu’en présence d’une dimension de Krull, ce type de section est la seule obstruction à la finitude du rang sans torsion. / In the fist part, we study the return probability of finitely generated metabelian groups. We give a characterization of such groups with large return probability in purely algebraic terms, namely the Krull dimension of the group. To do so, we establish, for metabelian groups, a variation of a famous embedding theorem of Kaloujinine and Krasner that respects this dimension. Along the way, we obtain lower and upper bounds on the return probability of metabelian groups according to their dimension.The second part of this thesis deals with isoperimetric profiles of locally compact compactly generated groups, that we use to characterize the existence of sequences of Følner couples. We generalize at a compact scale previous results of Tessera, in particular that they increase when going to a quotient group, so as to state in more generality a result from the first part, namely that the existence of Følner couples goes to a quotient group. We also prove that it goes to a closed subgroup. This allows to obtains a more self-contained proof of the main result of the first part of this thesis.The third part is a joint work with Kropholler in which we study the structure of soluble groups of infinite torsion-free rank with no ZwrZ. As a corollary, we obtain that a finitely generated soluble group with Krull dimension has finite torsion-free rank if and only if it has no ZwrZ.

We study the free centre-by-metabelian Lie ring, that is, the free Lie ring with the property that the second derived ideal is contained in the centre. We exhibit explicit generating sets for the homogeneous components and the fine homogeneous components of the second derived ideal. Each of these components is a direct sum of a free abelian group and a (possibly trivial) elementary abelian $2$-group. Our generating sets are such that some of their elements generate the torsion subgroup while the remaining ones freely generate a free abelian group. A key ingredient of our approach is the determination of the dimensions of the corresponding homogeneous components of the free centre-by-metabelian Lie algebra over fields of characteristic other than $2$. For this we exploit a $6$-term exact sequence of modules over a polynomial ring that is originally defined over the integers, but turns into a sequence whose terms are projective modules after tensoring with a suitable field. Our results correct a partly erroneous theorem in the literature. Moreover, we study the product of three homogeneous components of a free Lie algebra. Let $L$ be a free Lie algebra of finite rank over a field and let $L_n$ denote the degree $n$ homogeneous component of $L$. Formulae for the dimension of the subspaces $[L_n,L_m]$ for all $n$ and $m$ were obtained by Ralph St\"{o}hr and Michael Vaughan-Lee. Formulae for the dimension of the subspaces of the form $[L_n,L_m,L_k]$ under certain conditions on $n,m$ and $k$ were obtained by Nil Mansuro\u{g}lu and Ralph St\"{o}hr. Surprisingly, in contrast to the case of a product of two homogeneous components, the dimension of such products may depend on the characteristic of the field. For example, the dimension of $[L_2,L_2,L_1]$ over fields of characteristic $2$ is different from the dimension over fields of characteristic other than $2$.