On extremal properties of hyperbolic Coxeter polytopes and their reflection groups

This thesis concerns hyperbolic Coxeter polytopes, their reflection groups and associated combinatorial and geometric invariants. Given a Coxeter group $G$ realisable as a discrete subgroup of $\mathrm{Isom}\,\mathbb{H}^n$, there is a fundamental domain $\mathscr{P} \subset \mathbb{H}^n$ naturally associated to it. The domain $\mathscr{P}$ is a Coxeter polytope. Vice versa, given a Coxeter polytope $\mathscr{P}$, the set of reflections in its facets generates a Coxeter group acting on $\mathbb{H}^n$.

The reflections give a natural set $S$ of generators for the group $G$. Then we can express the growth series $f_{(G,S)}(t)$ of the group $G$ with respect to the generating set $S$. By a result of R.~Steinberg, the corresponding growth series is the power series of a rational function. The growth rate $\tau$ of $G$ is the reciprocal to the radius of convergence of such a series. The growth rate is an algebraic integer and, by a result of J.~Milnor, $\tau > 1$. By a result of W.~Parry, if $G$ acts on $\mathbb{H}^n$, $n=2,3$, cocompactly, then its growth rate is a Salem number. By a result of W.~Floyd, there is a geometric correspondence between the growth rates of cocompact and finite co-volume Coxeter groups acting on $\mathbb{H}^2$. This correspondence gives a geometric picture for the convergence of Salem numbers to Pisot numbers. There, Pisot numbers correspond to the growth rates of finite-volume polygons with ideal vertices. We reveal an analogous phenomenon in dimension $3$ by considering degenerations of compact Coxeter polytopes to finite-volume Coxeter polytopes with four-valent ideal vertices. In dimension $n\geq 4$, the growth rate of a Coxeter group $G$ acting cocompactly on $\mathbb{H}^n$ is known to be neither a Salem, nor a Pisot number.

A particularly interesting class of Coxeter groups are right-angled Coxeter groups. In case of a right-angled Coxeter group acting on $\mathbb{H}^n$, its fundamental domain $\mathscr{P} \subset \mathbb{H}^n$ is a right-angled polytope. Concerning the class of right-angled polytopes in $\mathbb{H}^4$ (compact, finite volume or ideal, as subclasses), the following questions emerge:

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\item[-] what are minimal volume polytopes in these families?

\item[-] what are polytopes with minimal number of combinatorial compounds (facets, faces, edges, vertices) in these families?

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The answer in the case of finite-volume right-angled polytopes is given by \`{E}.~Vinberg and L.~Potyaga\u{\i}lo. In the case of compact right-angled polytopes the answer is conjectured by the same authors. In this thesis, the above question in the case of ideal right-angled polytopes are considered and completely answered. We conclude with some partial results concerning the case of compact right-angled polytopes.