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On the residual finiteness of \(\text{Out}(\pi_1(M))\) of certain Seifert manifolds. (English) Zbl 1040.20023

It is shown that the outer automorphism group \(\text{Out}(\pi_1(M))\) of the fundamental group of a 3-dimensional Seifert fiber space \(M\) with non-empty boundary is residually finite. The proof uses a result of E. K. Grossman stating that \(\text{Out}(G)\) is residually finite for a group \(G\) which is conjugacy separable and has the property that an automorphism of \(G\) which maps each element to a conjugate is an inner automorphism; see [J. Lond. Math. Soc., II. Ser. 9, 160-164 (1974; Zbl 0292.20032)], where it is also shown that the outer automorphism group and mapping class group of a compact orientable surface is residually finite.

MSC:

20E26 Residual properties and generalizations; residually finite groups
20E36 Automorphisms of infinite groups
20F34 Fundamental groups and their automorphisms (group-theoretic aspects)
57M05 Fundamental group, presentations, free differential calculus
57M07 Topological methods in group theory

Citations:

Zbl 0292.20032
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