  Assume $G$ is a group and $\mathcal{A}$ is an algebra of subsets of $G$
closed under left translation. We study various ways to understand the Ellis
group of the $G$-flow $S(\mathcal{A})$ (the Stone space of $\mathcal{A}$), with
particular interest in the model-theoretic setting where $G$ is definable in a
first order structure $M$ and $\mathcal{A}$ consists of externally definable
subsets of $G$.
  In one part of the thesis we explore strongly generic sets. Maximal algebras
of such sets are shown to carry enough information to retrieve the Ellis group.
A subset of $G$ is strongly generic if each non-empty Boolean combination of
its translates is generic. Trivial examples include what we call *periodic*
sets, which are unions of cosets of finite index subgroups of $G$. We give
several characterizations of strongly generic sets, in particular, we relate
them to almost periodic points of the flow $2^G$. For groups without a smallest
finite index subgroup we show how to construct non-periodic strongly generic
subsets in a systematic way. When $G$ is definable in a model $M$, a definable,
strongly generic subset of $G$ will remain as such in any elementary extension
of $M$ only if it is strongly generic in $G$ in an adequately uniform way. Sets
satisfying this condition are called *uniformly strongly generic*. We analyse a
few examples of these sets in different groups.
  In the second part we assume that $G$ is a topological group and consider a
particular algebra of its subsets denoted $\mathcal{SBP}$. It consists of
subsets of $G$ that have the *strong Baire property*, meaning nowhere dense
boundary. We explicitly describe the Ellis group of $S(\mathcal{A})$ for an
arbitrary subalgebra $\mathcal{A}$ of $\mathcal{SBP}$ under varying assumptions
on the group $G$, including the case when $G$ is a compact topological group.
[...]
  (Full abstract in the article)
