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). / Mathematics
| Identifer | oai:union.ndltd.org:TEMPLE/oai:scholarshare.temple.edu:20.500.12613/8339 |
| Date | January 2022 |
| Creators | Choi, Benjamin Dongbin |
| Contributors | Stover, Matthew, Futer, David, Taylor, Samuel J., Meyer, Jeffrey |
| Publisher | Temple University. Libraries |
| Source Sets | Temple University |
| Language | English |
| Detected Language | English |
| Type | Thesis/Dissertation, Text |
| Format | 56 pages |
| Rights | IN COPYRIGHT- This Rights Statement can be used for an Item that is in copyright. Using this statement implies that the organization making this Item available has determined that the Item is in copyright and either is the rights-holder, has obtained permission from the rights-holder(s) to make their Work(s) available, or makes the Item available under an exception or limitation to copyright (including Fair Use) that entitles it to make the Item available., http://rightsstatements.org/vocab/InC/1.0/ |
| Relation | http://dx.doi.org/10.34944/dspace/8310, Theses and Dissertations |
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