The Euler characteristic of graph products and of Coxeter groups

On volumes of hyperbolic Coxeter polytopes and quadratic forms

“…where χ(P n ) is the Euler characteristic of the Coxeter group P n defined by P n . By Proposition 3 of Chiswell [3], we have that…”

“…Then ∆ 3 is arithmetic and Meyerhoff [12] proved that H 3 /∆ 3 has minimum volume among all orientable, noncompact, hyperbolic 3-orbifolds. The group ∆ 3 is the orientation preserving subgroup of a hyperbolic Coxeter group of type [3,3,6] [4,3,6]. Now the Coxeter tetrahedron of type [6, 3 1,1 ] can be subdivided into 5 copies of the Coxeter tetrahedron of type [3,3,6].…”

Weyl groups, lattices and geometric manifolds

“…: by writing the generating reflections s i as matrices in terms of the basis {x i } for the root lattice and proceeding from there to get im (ξ h/2 + 1) = x 1 + x 4 , x 2 + x 5 …”

Small volume closed hyperbolic 4-manifolds

“…The Euler characteristic of a group G is defined provided that G satisfies a cohomological finiteness condition. Coxeter groups satisfy this condition, and in [1] Chiswell proved that the Euler characteristic for a Coxeter group Γ can be computed in terms of the orders of parabolic subgroups of finite order. The Euler characteristic χ(H) of a subgroup H is given by χ(H) = χ(Γ).|Γ : H| and the Euler characteristic of the manifold H 4 /H is χ(H).…”