Linear slices close to a Maskit slice. (English) Zbl 1305.30020
Let \(S\) be a torus with one open disk removed. Put \(N=S\times [0,1 ]\) and \(P=\partial S \times [0,1]\). Let \(AH(N,P)\) be the space of faithful, type-preserving representations \(\rho: \pi_1(N) \to\mathrm{PSL}(2,\mathbb{C})\) with discrete images. In the paper under review, the author studies the topology of \(AH(N,P)\) near the Maskit slice by the trace coordinates. The author answers the question whether the linear slice \(L(\beta)\) converges to the Maskit slice \(L(2)\) as \(\beta\in \mathbb{C}\setminus [-2,2]\) tends to 2. The answer depends on whether a sequence of \(\beta_n\in \mathbb{C}\setminus [-2,2]\) converges to 2 horocyclically or tangentially. The theory of K. W. Bromberg [Duke Math. J. 156, No. 1, 387–427 (2011; Zbl 1213.30078)] on a local model of \(AH(N,P)\) plays an important role, so that the author describes a relation between Bromberg’s coordinates and trace coordinates for \(AH(N,P)\). The non-local connectivity of \(AH(N,P)\) is also considered and it is shown that there exists a linear slice \(L(\alpha)\) which is not locally connected at \(2\in \partial L(\alpha)\). The main results are also described in terms of complex Fenchel-Nielsen coordinates. The paper provides figures of linear slices generated by computer.
Reviewer: Gou Nakamura (Toyota)
MSC:
| 30F40 | Kleinian groups (aspects of compact Riemann surfaces and uniformization) |
| 57M50 | General geometric structures on low-dimensional manifolds |
Keywords:
Kleinian groups; once-punctured torus; quasifuchsian space; linear slices; Maskit slices; complex Fenchel-Nielsen coordinatesCitations:
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