In this work, we prove three types of results with the strategy that, together, the author believes these should imply the local version of Hilbert's Fifth problem. In a separate development, we construct a nontrivial topology for rings of map germs on Euclidean spaces. First, we develop a framework for the theory of (local) nonstandard Lie groups and within that framework prove a nonstandard result that implies that a family of local Lie groups that converge in a pointwise sense must then differentiability converge, up to coordinate change, to an analytic local Lie group, see corollary 6.3.1. The second result essentially says that a pair of mappings that almost satisfy the properties defining a local Lie group must have a local Lie group nearby, see proposition 7.2.1. Pairing the above two results, we get the principal standard consequence of the above work which can be roughly described as follows. If we have pointwise equicontinuous family of mapping pairs (potential local Euclidean topological group structures), pointwise approximating a (possibly differentiably unbounded) family of differentiable (sufficiently approximate) almost groups, then the original family has, after appropriate coordinate change, a local Lie group as a limit point. (See corollary 7.2.1 for the exact statement.) The third set of results give nonstandard renditions of equicontinuity criteria for families of differentiable functions, see theorem 9.1.1. These results are critical in the proofs of the principal results of this paper as well as the standard interpretations of the main results here. Following this material, we have a long chapter constructing a Hausdorff topology on the ring of real valued map germs on Euclidean space. This topology has good properties with respect to convergence and composition. See the detailed introduction to this chapter for the motivation and description of this topology.

Une partie symétrique X d'un groupe G est un sous-groupe K-approximatif s'il existe une partie finie E ⊂ G de taille K telle que X2 ⊂ E.X. L'étude combinatoire des groupes approximatifs a grandement bénéficié des apports de la Théorie des Modèles : en 2009, Hrushovski montre qu'une ultralimite de groupes approximatifs finis possède une composante connexe modèle-théorique, donc un quotient localement compact X/H. En appliquant les résultats de Gleason et Yamabe sur le cinquième problème de Hilbert, cela permet de trouver un morphisme vers un groupe de Lie, et d'en déduire des résultats de nilpotence. Cela a permis à Breuillard, Green et Tao de classifier tous les groupes approximatifs finis, en retrouvant un quotient X/H de manière combinatoire. Dans cette thèse, on s'intéresse à la construction d'un sous-groupe H type-définissable et d'indice borné, qui garantit l'existence d'un quotient localement compact. On montre que l'approche combinatoire de Breuillard, Green et Tao peut être vue de cette manière, et on la généralise à tous les groupes approximatifs définissablement moyennables. On montre aussi que si H est type-définissable dans un langage L∗, alors on peut construire un sous-groupe H qui est type-définissable sur un langage réduit L, et toujours d'indice borné. L'existence de H ne dépend donc pas du choix du langage / A symmetric subset X in a group G is a K-approximate subgroup if there exists a finite set E ⊂ G of cardinality K such that X2 ⊂ E.X. The study of approximate subgroups in multiplicative combinatorics experienced a significate advance through the use of model theory. In 2009, Hrushovski showed that an ultralimit of finite approximate subgroups has a model-theoretic connected component, thus a locally compact quotient X/H. Using the results of Gleason and Yamabe about Hilbert’s fifth problem, this allows the construction of a morphism to a Lie group, and deduce some results about nilpotency. This lead to the theorem of Breuillard, Green and Tao classifying all finite approximate subgroups, using a combinatorial construction of the quotient X/H. In this thesis, we are intersested in the conditions needed to construct a type definable subgroup H of bounded index in X. This implies the existence of a locally compact quotient.We show that the combinatorial construction of Breuillard, Green and Tao can be seen in a definable way, and give a generalisation to all definably amenable approximate subgroups. Also, we show that if H is type-definable in a language L∗, then it is possible to construct a subgroup H which is type-definable in a reduct L, still with bounded index. Thus the existence of a subgroup H does not depend on the choice of a base language.