Complex hyperbolic (3,3,n) triangle groups

Abstract

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Spherical CR uniformization of Dehn surgeries of the Whitehead link complement

“…Then the Dehn surgery of the Whitehead link complement on T 1 of slope 1 n−3 admits a spherical CR uniformization, given by the group Γ n . Once again, our proof only holds for n ≥ 9; and the use of the Poincaré polyhedron theorem is essentially the same as the one done by Parker, Wang and Xie in [PWX16]. Considering the results on the combinatorics of the intersections shown in [PWX16], we can complete the proof for the five last cases.…”

“…Once again, our proof only holds for n ≥ 9; and the use of the Poincaré polyhedron theorem is essentially the same as the one done by Parker, Wang and Xie in [PWX16]. Considering the results on the combinatorics of the intersections shown in [PWX16], we can complete the proof for the five last cases. For α 2 ∈ [ π 6 , α lim 2 [, the Poincaré polyhedron theorem can still be applied, and apart from the condition of being a polyhedron (where all faces are homeomorphic to balls), we check the hypothesis for α 2 ∈]0, π 6 [.…”

“…More recently, in [Der06], Deraux studies the representations of (4, 4, 4)-triangle groups, and shows that the representation for which the image of σ 1 σ 2 σ 1 σ 3 is of order 5 is a lattice in PU(2, 1). In [PWX16], Parker, Wang and Xie study the representations of (3, 3, n)-triangle groups, and prove Schwartz conjecture in this case. Namely, the discrete and faithful representations are the ones for which the image of σ 1 σ 2 σ 1 σ 3 is nonelliptic.…”

“…The parameters for which [U ] is of order 4, 5, 6, 7 or 8 are also suitable for the following, but our approach using visual spheres to control global intersections does not allow us to conclude for these parameters. However, the result is still true; a proof can be found in [PWX16], where Parker, Wang and Xie show, using other techniques and parametrizing some triangle groups, that the global combinatorics of the bisectors J ± k is the expected one when [U ] is elliptic of type ( 1 n , −1 n ) with n ≥ 4. We begin by fixing some notation and recalling the uniformization result of Parker and Will, which gives a spherical CR structure on W LC for the parameter α 2 = α lim 2 .…”

“…The techniques used for the proof do not let us treat the last five cases. However, the global combinatorics of bisectors that would be needed to conclude for n ≥ 4 are shown by Parker, Wang and Xie in [PWX16], but with other techniques, using in particular a parametrization of a family of triangle groups. The result on the global combinatorics corresponds to the statement of Theorem 4.3 in [PWX16]; the link between their notation and ours is given by U = I 1 I 2 , S −1 = I 1 I 3 and T = I 3 I 2 .…”