ONE-CUSPED CONGRUENCE SUBGROUPS OF SO(d, 1; Z)

Abstract

The classical spherical and Euclidean geometries are easy to visualize and correspond to spaces with constant curvature 0 and +1 respectively. The geometry with constant curvature −1, hyperbolic geometry, is much more complex. A powerful theorem of Mostow and Prasad states that in all dimensions at least 3, the geometry of a finite-volume hyperbolic manifold (a space with local d-dimensional hyperbolic geometry) is determined by the manifold's fundamental group (a topological invariant of the manifold). A cusp is a part of a finite-volume hyperbolic manifold that is infinite but has finite volume (cf. the surface of revolution of a tractrix has finite area but is infinite). All non-compact hyperbolic manifolds have cusps, but only finitely many of them. In the fundamental group of such a manifold, each cusp corresponds to a cusp subgroup, and each cusp subgroup is associated to a point on the boundary of H^d, which can be identified with the (d − 1)-sphere. It is known that there are many one-cusped two- and three-dimensional hyperbolic manifolds. This thesis studies restrictions on the existence of 1-cusped hyperbolic d-dimensional manifolds for d ≥ 3. Congruence subgroups belong to a special class of hyperbolic manifolds called arithmetic manifolds. Much is known about arithmetic hyperbolic 3- manifolds, but less is known about arithmetic hyperbolic manifolds of higher dimensions. An important infinite class of arithmetic d-manifolds is obtained using SO(n, 1; Z), a subset of the integer matrices with determinant 1. This is known to produce 1-cusped examples for small d. Taking special congruence conditions modulo a fixed number, we obtain congruence subgroups of SO(n, 1; Z) which also have cusps but possibly more than one. We ask what congruence subgroups with one cusp exist in SO(n, 1; Z). We consider the prime congruence level case, then generalize to arbitrary levels. Covering space theory implies a relation between the number of cusps and the image of a cusp in the mod p reduced group SO(d+ 1, p), an analogue of the classical rotation Lie group. We use the sizes of maximal subgroups of groups SO(d + 1, p), and the maximal subgroups' geometric actions on finite vector spaces, to bound the number of cusps from below. Let Ω(d, 1; Z) be the index 2 subgroup in SO(d, 1; Z) that consists of all elements of SO(d, 1; Z) with spinor norm +1. We show that for d = 5 and d ≥ 7 and all q not a power of 2, there is no 1-cusped level-q congruence subgroup of Ω(d, 1; Z). For d = 4, 6 and all q not of the form 2^a3^b, there is no 1-cusped level-q congruence subgroup of Ω(d, 1; Z).