  Breaking symmetries is a popular way of speeding up the branch-and-bound
method for symmetric integer programs. We study fundamental domains, which are
minimal and closed symmetry breaking polyhedra. Our long-term goal is to
understand the relationship between the complexity of such polyhedra and their
symmetry breaking capability.
  Borrowing ideas from geometric group theory, we provide structural properties
that relate the action of the group with the geometry of the facets of
fundamental domains. Inspired by these insights, we provide a new generalized
construction for fundamental domains, which we call generalized Dirichlet
domain (GDD). Our construction is recursive and exploits the coset
decomposition of the subgroups that fix given vectors in $\mathbb{R}^n$. We use
this construction to analyze a recently introduced set of symmetry breaking
inequalities by Salvagnin (2018) and Liberti and Ostrowski (2014), called
Schreier-Sims inequalities. In particular, this shows that every permutation
group admits a fundamental domain with less than $n$ facets. We also show that
this bound is tight.
  Finally, we prove that the Schreier-Sims inequalities can contain an
exponential number of isomorphic binary vectors for a given permutation group
$G$, which provides evidence of the lack of symmetry breaking effectiveness of
this fundamental domain. Conversely, a suitably constructed GDD for this $G$
has linearly many inequalities and contains unique representatives for
isomorphic binary vectors.
