Finiteness theorems for congruence reflection groups

Abstract

This paper is a follow-up to the paper I. Agol, M. Belolipetsky, P. Storm, K. Whyte, Finiteness of arithmetic hyperbolic reflection groups, Groups, Geometry, and Dynamics 2 (2008), 481-498. The main purpose is to investigate the effective side of the method developed there and its possible application to the problem of classification of arithmetic hyperbolic reflection groups.

Arithmetic hyperbolic reflection groups

“…Now let us assume that Γ is at the same time a congruence subgroup and a reflection group, and let O = H n /Γ. Following the argument in [Bel11], we can then quickly prove the two principal finiteness theorems. We have:…”

Untitled

“…By assumption there exists a ∈ F × such that aq 1 and q 2 are isometric. By part (1), it suffices to show that aq 1 and q 1 are isogroupic. Pick…”

Dirichlet–Ford domains and arithmetic reflection groups

“…Every dihedral angle of P is π 2 . The quotient space H 3 /Γ has 14 boundary components; the cusp at ∞ gives a boundary torus, and each of the 13 cusp cycles in C gives a (2, 2, 2, 2) or a (2,4,4) sphere. Thus the peripheral homology has rank 1.…”