New York Journal of Mathematics

New York J. Math. 8 (2002) 257–258.

Lifting Möbius Groups: Addendum

J. O. Button

Abstract. We show that if Γ is a group with H^{2}(Γ; ℤ_{2}) = 0 then every representation of Γ into PSL(2, ℂ) lifts to SL(2, ℂ), but the converse does not hold.

In [2] we consider the natural surjection π : SL(2, ℂ) → PSL(2, ℂ) and we examine the question of when a subgroup of PSL(2, ℂ) can be lifted to SL(2, ℂ). In Section 3 of that paper we let Γ be an abstract group and ask when a representation ρ : Γ → PSL(2, ℂ) can be lifted to a representation ρ : Γ → SL(2, ℂ); that is ρ = πρ. It is pointed out there that if ρ is a faithful representation then the question is the same as whether the subgroup ρ(Γ) has a lift, but it is not the same in general. A question that has been raised on this topic is which groups Γ have the property that every representation ρ ∈ Hom (Γ, PSL(2, ℂ)) can be lifted; certainly any free group possesses this property.

We first show the following proposition which is known and is outlined in [3, p. 118], [4, p. 174] and [1, p. 755].

Proposition 1. If H^{2}(Γ; ℤ_{2}) = 0 then every representation ρ of Γ into PSL(2, ℂ) can be lifted to SL(2, ℂ).

Proof. Given such a ρ, we seek a group Γ̂ with Γ̂/ℤ_{2} = Γ and, if ϕ : Γ̂ → Γ is the natural surjection, a representation ρ̂: Γ̂ → SL(2, ℂ) with πρ̂ = ρϕ. If so then Γ̂ is a central extension of ℤ_{2} by Γ, although note that (Γ̂, ϕ, ρ̂) is not uniquely determined by these conditions: if ρ does lift to ρ and ϕ : Γ̂ → Γ is any central extension of ℤ_{2} by Γ then setting ρ̂ = ρϕ will do. In our case we define Γ̂ using the pullback; this means that

Γ̂ = {(γ, x) : γ ∈ Γ, x ∈ π^{−1}(ρ(γ))},

ϕ(γ, x) = γ and ρ̂ is the homomorphism from Γ̂ to SL(2, ℂ) defined by ρ̂(γ, x) = x. Then if ϕ : Γ̂ → Γ is the trivial central extension, so that Γ̂ ≅ Γ × ℤ_{2} and we can define the homomorphism ϕ^{−1} : Γ → Γ×{I}, we get a lift of ρ by setting ρ = ρ̂ϕ^{−1}. Moreover if ρ can be lifted to ρ then the function (γ, x) ↦ x(ρ(γ))^{−1} is a surjective homomorphism from Γ̂ to ±I with its kernel projecting down under ϕ to Γ, thus Γ̂ must be the trivial central extension of Γ.

257^{ISSN 1076-9803/02}

Received December 16, 2002.

Mathematics Subject Classification. 30F40, 20H10.

Key words and phrases. lift, group.

However the central extensions of ℤ_{2} by Γ are described by the group H^{2}(Γ; ℤ_{2}) and so if H^{2}(Γ; ℤ_{2}) = 0 then there is only the trivial central extension which means that from above every representation of Γ into PSL(2, ℂ) must lift. □

A good source to look for examples of groups Γ with H^{2}(Γ; ℤ_{2}) = 0 is the fundamental group π_{1}X of an aspherical 3-manifold X, in which case the (co)homology groups of X are the same as those of π_{1}X. We know for M a compact aspherical 3-manifold with boundary that

H^{2}(π_{1}M; ℤ_{2}) = H^{2}(M; ℤ_{2}) = H_{1}(M, ∂M; ℤ_{2})

by Poincaré duality. If such a 3-manifold has one boundary component that is a torus and H_{1}(M; ℤ) = ℤ + G for G odd and finite then H_{1}(M, ∂M; ℤ_{2}) = 0 and so all representations can be lifted; included in these are all knot complements in S^{3}. If M is a closed aspherical 3-manifold then H^{2}(π_{1}M; ℤ_{2}) = H_{1}(M; ℤ_{2}) which is 0 if and only if H_{1}(M; ℤ) is finite and has odd order. We know of one example: the Siefert-Wieber dodecahedral space is hyperbolic, thus aspherical, and has H_{1}(M; ℤ) = (ℤ_{5})^{3} (see [5]) so every representation of its fundamental group lifts.

This begs the question: if all representations of Γ lift then is H^{2}(Γ; ℤ_{2}) = 0? The obvious candidates for counterexamples are simple groups that do not embed in PSL(2, ℂ) so that the only representation is the identity.

Proposition 2. The alternating groups A_{n} for n ≥ 8 have every PSL(2, ℂ)-repre-sentation liftable but H^{2}(A_{n}; ℤ_{2}) ≠ 0.

Proof. For n ≥ 5 we know that A_{n} is simple, and although A_{5} is a subgroup of PSL(2, ℂ) so that its faithful representation cannot be lifted, A_{n} is certainly not a subgroup of PSL(2, ℂ) for n ≥ 6 so that here only the identity representation exists which certainly can be lifted. The Schur multiplier of A_{n} is ℤ_{2} for n ≥ 8, which means that there exists a non-trivial central extension of ℤ_{2} by A_{n}. Thus H^{2}(A_{n}; ℤ_{2}) ≠ 0 but every representation lifts. □

References

[1] S. Boyer and X. Zhang, On Culler-Shalen seminorms and Dehn filling, Ann. of Math. (2) 148 (1998), 737–801, MR 2000d:57028.

[2] J. O. Button, Lifting Möbius groups, New York J. Math. 8 (2002), 181–188.

[3] D. Cooper and D. D. Long, An undetected slope in a knot manifold, Topology ’90 (Columbus, OH, 1990), de Gruyter, Berlin, 1992, 111–121, MR 93i:57022, Zbl 0771.57006.

[4] H. M. Hilden, M. T. Lozano and J. M. Montesinos-Amilibia, The arithmeticity of the figure eight knot orbifolds, Topology ’90 (Columbus, OH, 1990), de Gruyter, Berlin, 1992, 169–183, MR 93k:57012, Zbl 0767.57004.

[5] J. Hempel, Orientation reversing involutions and the first Betti number for finite coverings of 3-manifolds, Invent. Math. 67 (1982), 133–142, MR 83m:57010, Zbl 0494.57003.

Selwyn College, Cambridge, CB3 9DQ, U.K.

jb128@dpmms.cam.ac.uk http://www.dpmms.cam.ac.uk/site2000/Staff/button01.html

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