Pseudo-Anosov subgroups of general fibered 3-manifold groups. (English) Zbl 1535.57018
Let \(S\) be a finite type orientable surface of negative Euler characteristic and let \(f \colon S \to S\) be a reducible, infinite order mapping class. Let \(\Gamma_f\) denote the fundamental group of the mapping class group \(M_f = S \times [0,1] / (x, 0) \sim (f(x), 1)\) with a basepoint \(z \in M_f\). Let \(S^{z} = S \setminus \{z\}\), then via the Birman Exact Sequence, \(\Gamma_f\) is identified with a subgroup of \(\mathrm{Mod}(S^{z}, z) < \mathrm{Mod}(S^z)\) – the mapping class group of \(S\) with a marked point.
In this paper, the authors show that a subgroup \(G\) of \(\Gamma_f\) is convex cocompact if and only if it is finitely generated and purely pseudo-Anosov. Combined with results of S. Dowdall et al. [Groups Geom. Dyn. 8, No. 4, 1247–1282 (2014; Zbl 1321.57001)] and R. P. Kent IV et al. [J. Reine Angew. Math. 637, 1–21 (2009; Zbl 1190.57014)], this establishes the result for the image of all such fibered 3-manifold groups in the mapping class group.
In this paper, the authors show that a subgroup \(G\) of \(\Gamma_f\) is convex cocompact if and only if it is finitely generated and purely pseudo-Anosov. Combined with results of S. Dowdall et al. [Groups Geom. Dyn. 8, No. 4, 1247–1282 (2014; Zbl 1321.57001)] and R. P. Kent IV et al. [J. Reine Angew. Math. 637, 1–21 (2009; Zbl 1190.57014)], this establishes the result for the image of all such fibered 3-manifold groups in the mapping class group.
Reviewer: Hoang Thanh Nguyen (Đà Nẵng)
MSC:
| 57K20 | 2-dimensional topology (including mapping class groups of surfaces, Teichmüller theory, curve complexes, etc.) |
| 20F65 | Geometric group theory |
| 57K30 | General topology of 3-manifolds |
| 20E08 | Groups acting on trees |
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