Normal automorphisms of relatively hyperbolic groups. (English) Zbl 1227.20041
An automorphism \(\alpha\) of a group \(G\) is normal if \(\alpha(N)=N\) for every normal subgroup \(N\) of \(G\). Obviously every inner automorphism is normal. Let \(\operatorname{Aut}_n(G)\), \(\text{Inn}(G)\), \(\text{Out}_n(G)\) denote, respectively, the subset of normal automorphisms, the set of inner automorphisms and the quotient group \(\operatorname{Aut}_n(G)/\text{Inn}(G)\).
The authors prove that for any relatively hyperbolic group \(G\), \(\text{Out}_n(G)\) is finite, and if in addition \(G\) is non-elementary and has no finite non-trivial normal subgroups, then \(\text{Out}_n(G)=1\).
Earlier, similar results were known for many other classes of groups. For example, \(\text{Out}_n(G)\) is trivial for non-Abelian free groups (Lubotzky), for non-trivial free products (M. V. Neshchadim), fundamental groups of closed surfaces of negative Euler characteristic (O. Bogopolski, E. Kudryavtseva, H. Zieschang), non-Abelian free Burnside groups of large odd exponent (E. Cherepanov), non-Abelian free solvable groups (V. Roman’kov), free nilpotent groups of class \(c=2\) (for \(c\geq 3\) this is not true) (G. Endimioni), free profinite groups (Jarden), and free soluble pro-\(p\)-groups (N. Romanovskiĭ).
As an application of the main result, the authors prove that \(\text{Out}(G)=\operatorname{Aut}(G)/\text{Inn}(G)\) is residually finite for every finitely generated residually finite group \(G\) with infinitely many ends.
The authors prove that for any relatively hyperbolic group \(G\), \(\text{Out}_n(G)\) is finite, and if in addition \(G\) is non-elementary and has no finite non-trivial normal subgroups, then \(\text{Out}_n(G)=1\).
Earlier, similar results were known for many other classes of groups. For example, \(\text{Out}_n(G)\) is trivial for non-Abelian free groups (Lubotzky), for non-trivial free products (M. V. Neshchadim), fundamental groups of closed surfaces of negative Euler characteristic (O. Bogopolski, E. Kudryavtseva, H. Zieschang), non-Abelian free Burnside groups of large odd exponent (E. Cherepanov), non-Abelian free solvable groups (V. Roman’kov), free nilpotent groups of class \(c=2\) (for \(c\geq 3\) this is not true) (G. Endimioni), free profinite groups (Jarden), and free soluble pro-\(p\)-groups (N. Romanovskiĭ).
As an application of the main result, the authors prove that \(\text{Out}(G)=\operatorname{Aut}(G)/\text{Inn}(G)\) is residually finite for every finitely generated residually finite group \(G\) with infinitely many ends.
Reviewer: Natalia Yu. Makarenko (Mulhouse)
MathOverflow Questions:
Solving equations in hyperbolic groups and subgroups of isometry of a Gromov hyperbolic spaceMSC:
| 20F67 | Hyperbolic groups and nonpositively curved groups |
| 20E36 | Automorphisms of infinite groups |
| 20F28 | Automorphism groups of groups |
| 20E26 | Residual properties and generalizations; residually finite groups |
| 57M07 | Topological methods in group theory |
| 20F65 | Geometric group theory |
Keywords:
relatively hyperbolic groups; normal automorphisms; outer automorphism groups; groups with infinitely many ends; residual finiteness; group-theoretic Dehn surgeryReferences:
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