A composition formula in the rank two free group

González-Acuña, Francisco; Ramírez, Arturo

doi:10.1090/S0002-9939-99-05344-7

A composition formula in the rank two free group
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by Francisco González-Acuña and Arturo Ramírez PDF

Proc. Amer. Math. Soc. 127 (1999), 2779-2782 Request permission

Abstract:

Using the fundamental group of a punctured torus, a free group $F$ of rank two, and the fact that the natural eipmorphism from $Aut F$ onto $Aut(F/F’)$ has as kernel the group of inner automorphisms of $F$, we describe representatives of the conjugacy classes of generating pairs of $F$ and give explicit relations between them.

References

M. Cohen, Wolfgang Metzler, and A. Zimmermann, What does a basis of $F(a,\,b)$ look like?, Math. Ann. 257 (1981), no. 4, 435–445. MR 639577, DOI 10.1007/BF01465865
M. Lustig, E.-M. Thiele, and H. Zieschang, Primitive elements in the free product of two finite cyclic groups, Abh. Math. Sem. Univ. Hamburg 65 (1995), 277–281. MR 1359136, DOI 10.1007/BF02953334
Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89, Springer-Verlag, Berlin-New York, 1977. MR 0577064
J. Nielsen, Die Isomorphismen der allgemeinen, unendlichen Gruppe mit zwei Erzeugenden, Math. Ann. 78 (1918), 385-397.
R. P. Osborne and H. Zieschang, Primitives in the free group on two generators, Invent. Math. 63 (1981), no. 1, 17–24. MR 608526, DOI 10.1007/BF01389191
K. D. Stroyan and W. A. J. Luxemburg, Introduction to the theory of infinitesimals, Pure and Applied Mathematics, No. 72, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. MR 0491163