Finite groups of outer automorphisms of free groups

1992 ◽  
Vol 32 (5) ◽  
pp. 796-811 ◽  
Author(s):  
V. D. Mazurov
Keyword(s):  
Finite Groups ◽  
Free Groups ◽  
Automorphisms Of Free Groups ◽  
Outer Automorphisms

On representations of Artin–Tits and surface braid groups

2011 ◽  
Vol 14 (1) ◽  
Author(s):  
Valeriy G. Bardakov ◽  
Paolo Bellingeri
Keyword(s):  
Free Groups ◽  
Braid Groups ◽  
Finitely Generated ◽  
Induced Representations ◽  
Automorphisms Of Free Groups ◽  
Closed Surfaces ◽  
Outer Automorphisms

AbstractWe define and study extensions of the Artin and Perron–Vannier representations of braid groups to topological and algebraic generalizations of braid groups. We provide faithful representations of braid groups of oriented surfaces with boundary as automorphisms of finitely generated free groups. The induced representations of such groups as outer automorphisms of finitely generated free groups are still faithful. Also we give a representation of braid groups of closed surfaces as outer automorphisms of finitely generated free groups. Finally, we provide faithful representations of Artin–Tits groups of type 𝒟 as automorphisms of free groups.

scholarly journals Finite groups of outer automorphisms of free groups

1996 ◽  
Vol 38 (3) ◽  
pp. 275-282 ◽  
Author(s):  
Bruno Zimmermann
Keyword(s):  
Automorphism Group ◽  
Finite Groups ◽  
Outer Automorphism ◽  
Finite Subgroup ◽  
Canonical Projection ◽  
Model Case ◽  
Automorphisms Of Free Groups ◽  
Outer Automorphisms ◽  
Finite Subgroups ◽  
Inner Automorphisms

Let Fr denote the free group of rank r and Out Fr: = AutFr/Inn Fr the outer automorphism group of Fr (automorphisms modulo inner automorphisms). In [10] we determined the maximal order 2rr! (for r > 2) for finite subgroups of Out Fr as well as the finite subgroup of that order which, for r > 3, is unique up to conjugation. In the present paper we determine all maximal finite subgroups (that is not contained in a larger finite subgroup) of Out F3, up to conjugation (Theorem 2 in Section 3). Here the considered case r = 3 serves as a model case: our method can be applied for other small values of r (in principle for any value of r) but the computations become considerably longer and are more apt for a computer then; the method can also be applied to determine the maximal finite subgroups of the automorphism group Aut Fr of Fr. Note that the canonical projection Aut Fr ⃗ Out Fr is injective on finite subgroups of Aut Fr; however, not all finite subgroups of Out Fr lift to finite subgroups of Aut Fr.

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