The size of a hyperbolic Coxeter simplex

Hyperbolic Coxeter Pyramids

“…These have been well studied (c.f. [3][4][5][6] Tumarkin: in [7]; and in [8]. These lists are complete, and contain in their number many examples of hyperbolic Coxeter polytopes with the combinatorial type of a pyramid.…”

“…Note that the placement of the parabolic graph 4 constrains the labelling of the edges connecting the vertex v to the rest of the graph such that they must all be of weight 3. n   Assume that P has finite volume, and that . Then 4 is a connected component of …”

On commensurable hyperbolic Coxeter groups

“…We assemble the different symbols into a single one in order to describe the different nature of parts of the Coxeter graph in question (see Fig. 1 and also [13]). …”

“…In particular, the hyperbolic Coxeter simplex groups, characterised by N = n + 1, were classified by Lannér in the cocompact case and by Koszul in the non-cocompact case; they exist up to n = 9. In [13,14], the covolumes and commensurability classes of all cofinite hyperbolic simplex groups were determined.…”

Hyperbolic Orbifolds of Minimal Volume

“…The volumes of all hyperbolic Coxeter simplices (existing in H n for n ≤ 9) have been calculated in [18] by the application of several methods, ranging from analytical, combinatorial to arithmetic techniques. In Table 1, we present the minimal (and some …”