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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.