 # Normal Congruence Subgroups of the Bianchi Groups and Related Groups

Scarth, Robert M (1999) Normal Congruence Subgroups of the Bianchi Groups and Related Groups. PhD thesis, University of Glasgow.

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## Abstract

Let O be an order in an imaginary quadratic number field. This thesis is mainly concerned with normal subgroups of SL2(O) and of PSL2(O). Suppose that O is a maximal order then O is the ring of integers in the number field and the group PSL2(O) is a Bianchi group. In chapter one we discuss the geometric background of these groups and introduce some fundamental algebraic concepts; those of order and level. We also discuss the Congruence subgroup problem. Chapter two is a discussion of the fundamental theorem of Zimmert . In chapter three we discuss PSL2(O) where O is not a maximal order. We derive a formula for their index in the Bianchi groups and presentations for some of these groups. In particular we derive a presentation for PSL2 (Z[√-3])and using this presentation get a partial classification of the normal subgroups of PSL2(Z[√-3]). Chapter four generalizes a result of Mason and Pride  about SL2(Z) to all but finitely many SL2(O). This result shows that for an arbitrary normal subgroup of N < SL2(O) there is no relationship between the order and level of N. This is in distinction to the groups SLn(O), n > 3, where the order and level of a normal subgroup coincide. This answers a question of Lubotzky's. Let O be an order in an imaginary quadratic number field. Then O is a Noetherian domain of Krull dimension one and has characteristic zero. Chapter five discusses SL2 over the class of all Noetherian domains of Krull dimension one, including those of nonzero characteristic. In particular we generalize the work of Mason  and derive a relationship between the order and level of a normal congruence subgroup of SL2(K) for any Noetherian domain of Krull dimension one, K. In chapter six we apply this work to SL2(O) and construct a new and vast class of normal non-congruence subgroups of SL2(O). Finally we take a closer look at some particular PSL2(O).

Item Type: Thesis (PhD) Doctoral Adviser: Alec Mason Mathematics 1999 Enlighten Team glathesis:1999-76467 Copyright of this thesis is held by the author. 19 Nov 2019 14:18 19 Nov 2019 14:18 http://theses.gla.ac.uk/id/eprint/76467 View Item