Reflection subgroups of Euclidean reflection groups
In this paper we classify reflection subgroups of Euclidean Coxeter groups.
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arXiv subjects
Pavel Tumarkin
Publications and source records attributed to Pavel Tumarkin.
On the volume of a six-dimensional polytope
This note is a comment to the paper by D.R.Heath-Brown and B.Z.Moroz (Math Proc. Camb. Phil. Soc. 125 (1999)). That paper concerns with the projective surface $S$ in $\mathbb{P}^{3}$ defined by the equation $x_{1}x_{2}x_{3}=x_{4}^{3}$. It is shown there that the evaluation of the leading term of the asymptotic formula for the number of rational points of bounded height in $S(\Q)$ is equivalent to the evaluation of the volume of some 6-dimensional polytope $¶$. The volume of $¶$ is known from several papers; we calculate this volume by elementary method using symmetry of the polytope $¶$. We also discuss a combinatorial structure of $¶$.
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Euclidean simplices generating discrete reflection groups
Let $P$ be a convex polytope in the Euclidean space $\E^n$. Consider the group $G_P$ generated by reflections in the facets of $P$. We say that $P$ {\it generates a reflection group $G_P$}. In the present paper, we list all Euclidean simplices generating discrete reflection groups.
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Hyperbolic Coxeter n-polytopes with n+3 facets
A polytope is called a Coxeter polytope if its dihedral angles are integer parts of $π$. In this paper we prove that if a non-compact Coxeter polytope of finite volume in $H^n$ has exactly $n+3$ facets then $n\le 16$. We also find an example in $H^{16}$ and show that it is unique.
math.MG↗