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Normal closures of powers of Dehn twists in mapping class groups. (English) Zbl 0760.57005

Let \(F=F(g,n)\) be an oriented surface of genus \(g\geq 1\) with \(n<2\) boundary components and let \(M(F)\) be its mapping class group. Let \(e\) be a non-bounding simple closed curve in \(F\) and let \(E\) denote the isotopy class of the Dehn twist about \(e\). Let \(N\) be the normal closure of \(E^ 2\) in \(M(F)\). In this paper we answer a question of J. S. Birman [Braids, links, and mapping class groups (1975; Zbl 0305.57013), p. 219]:
Theorem 1. The subgroup \(N\) is of finite index in \(M(F)\).
In fact we prove somewhat more:
Theorem 2. If \(F\) is closed and has genus two or three, then the normal closure of \(E^ 3\) is of finite index in \(M(F)\).
Theorem 3. If \(F\) has genus two and has a single boundary component, then the normal closure of \(E^ 2\) or \(E^ 3\) is of finite index in \(M(F)\).
On the other hand we prove:
Theorem 4. If \(F\) has genus two and has \(n\geq 0\) boundary components, then the normal closure of \(E^ k\) is of infinite index in \(M(F)\) for all \(k>3\).

MSC:

57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
57R50 Differential topological aspects of diffeomorphisms
57M60 Group actions on manifolds and cell complexes in low dimensions

Citations:

Zbl 0305.57013
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References:

[1] DOI: 10.1016/0040-9383(85)90049-7 · Zbl 0571.57009 · doi:10.1016/0040-9383(85)90049-7
[2] Johnson, I 249 pp 225– (1980)
[3] DOI: 10.2307/1971403 · Zbl 0631.57005 · doi:10.2307/1971403
[4] Birman, Braids, links and mapping class groups (1975) · Zbl 0305.57013 · doi:10.1515/9781400881420
[5] DOI: 10.2307/2043120 · Zbl 0391.57009 · doi:10.2307/2043120
[6] DOI: 10.1007/BF01360285 · Zbl 0134.26502 · doi:10.1007/BF01360285
[7] Magnus, Combinatorial group theory (1976)
[8] DOI: 10.1017/S030500410003824X · doi:10.1017/S030500410003824X
[9] DOI: 10.2307/2006977 · Zbl 0549.57006 · doi:10.2307/2006977
[10] Newman, Integral matrices (1972)
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