Subgroups of some (2, 3, n) triangle groups

[摘要] As an abstract group, the (2,3,n) triangle group has the presentation mit_n = < x,y : x^2 = y^3 = (yx)^n = 1 > This thesis is concerned with subgroups of finite index in mit9, mit_11 and mit 13. With a subgroup of finite index, u, in the (2,3,11) triangle group, we associate a quintuple of non-negative integers (u,p,e,f,g), with u1 and 5u = 132(p - 1) + 33e + 44f + 60g. We show in Theorem 1.4.6 that each quintuple, satisfying the conditions, corresponds to a subgroup of mit11. With a subgroup of finite index, u, in the (2,3,12) triangle group, we associate a quintuple of non-negative integers (u,p,e,f,g), with u1 and 7u = 156(p - 1) + 39e + 52f + 72g. We show in Theorem 3.3.6 that each quintuple, satisfying the conditions, corresponds to a subgroup of mit13. With a subgroup of finite index, u, in the (2,3,9) triangle group, we associate a sextuple of non-negative integers (u,p,e,f,g1,g3) with u1, u = f (mod 3) and u = 36(p - 1) + 9e + 12f + 16g_1 + 12g_3. We show in Theorem 2,3,9 that each sextuple, satisfying the conditions, corresponds to a subgroup of mit9 with the following exceptions: (a) (12n+ 9,0,1,0,0,n+ 3), V n0 (b) (24,0,0,0,0,5) (c) (24,0,0,0,3,1) (d) (24,0,0,3,0,2) Coset diagrams are used extensively in the proofs, although to prove exception (a) for mit9, we make use of Hauptmodul equations (see [1] and [23]). Computer programs were developed to generate all quintuples satisfying the relevant conditions for (2,3,110 subgroups for u101, all quintuples satisfying the relevant conditions for (2,3,13) subgroups for u110, and all sextuples satisfying the relevant conditions for (2,3,9) subgroups for u38. These programs and their output are presented in the Appendices. We show in Theorem 1.2.2 that quintuples, which satisfy the relevant (2,3,11) conditions, exist for each u99. We show in Theorem 2.2.1 that sextuples, which satisfy the relevant (2,3,9) conditions, exist for each u36. We show in Theorem 3.2.1 that quintuples, which satisfy the relevant (2,3,13) conditions, exist for each u104.

[发布机构] University:University of Glasgow;Department:School of Mathematics and Statistics

[关键词] QA Mathematics