One-cusped congruence subgroups of Bianchi groups

Petersen, Kathleen L.

doi:10.1007/s00208-006-0065-z

Cited by 6 publications

(5 citation statements)

References 16 publications

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“…There are one-cusped hyperbolic manifolds commensurable with the modular curve, and, though the figure-eight knot is the unique arithmetic knot complement [14], several Bianchi groups PSL 2 (O d ) contain a one-cusped manifold. See also [9]. The corresponding question for higher dimensional hyperbolic manifolds also appears to be unsolved.…”

Section: Question Does There Exist a One-cusped Complex Hyperbolic 2mentioning

confidence: 99%

Cusps of Picard modular surfaces

Stover

2011

Geom Dedicata

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Section: Question Does There Exist a One-cusped Complex Hyperbolic 2mentioning

confidence: 99%

Cusps of Picard modular surfaces

Stover

2011

Geom Dedicata

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“…This paper is an extension of part of the author's doctoral thesis [6]. The author would like to thank her advisor, Alan Reid, for his guidance and support.…”

Section: Acknowledgementsmentioning

confidence: 99%

“…Later, Petersson [9] proved that there are only finitely many one-cusped congruence subgroups of the modular group, and that the indices of such groups are the divisors of 55440 = 11 · 7 · 5 · 3 2 · 2 4 . The commutator subgroup of PSL 2 (Z), a subgroup of index 6, is a torsion-free one-cusped congruence subgroup containing Γ (6).…”

Section: Introductionmentioning

confidence: 99%

Counting cusps of subgroups of $\mathrm {PSL}_2(\mathcal {O}_K)$

Petersen

2008

Proc. Amer. Math. Soc.

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Abstract. Let K be a number field with r real places and s complex places, and let O K be the ring of integers of K.has h K cusps, where h K is the class number of K. We show that under the assumption of the generalized Riemann hypothesis that if K is not Q or an imaginary quadratic field and if i ∈ K, then PSL 2 (O K ) has infinitely many maximal subgroups with h K cusps. A key element in the proof is a connection to Artin's Primitive Root Conjecture.

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“…It is unknown to the author whether or not there exist 1-cusped Hilbert modular manifolds. Using the generalized Riemann hypothesis, K. Petersen [11] constructed infinite many 1-cusped Hilbert modular surfaces. However, the nature of the construction likely produces Hilbert modular surface groups with 2-torsion.…”

Section: Remarkmentioning

confidence: 99%

Cusps of Hilbert modular varieties

McReynolds¹

2008

Math. Proc. Camb. Phil. Soc.

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Motivated by a question of Hirzebruch on the possible topological types of cusp cross-sections of Hilbert modular varieties, we give a necessary and sufficient condition for a manifold M to be diffeomorphic to a cusp cross-section of a Hilbert modular variety. Specialized to Hilbert modular surfaces, this proves that every Sol 3-manifold is diffeomorphic to a cusp cross-section of a (generalized) Hilbert modular surface. We also deduce an obstruction to geometric bounding in this setting. Consequently, there exist Sol 3-manifolds that cannot arise as a cusp crosssection of a 1-cusped nonsingular Hilbert modular surface.

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One-cusped congruence subgroups of Bianchi groups

Abstract: We show that there are only finitely many maximal congruence subgroups of the Bianchi groups such that the quotient by H 3 has only one cusp.

Cited by 6 publications

References 16 publications

Cusps of Picard modular surfaces

Cusps of Picard modular surfaces

Counting cusps of subgroups of $\mathrm {PSL}_2(\mathcal {O}_K)$

Cusps of Hilbert modular varieties

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