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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

An inverse function theorem for free groups


Author: Joan S. Birman
Journal: Proc. Amer. Math. Soc. 41 (1973), 634-638
MSC: Primary 20E05
DOI: https://doi.org/10.1090/S0002-9939-1973-0330295-8
MathSciNet review: 0330295
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Abstract: Let ${F_n}$ be a free group of rank $n$ with free basis ${x_1}, \cdots ,{x_n}$. Let $\{ {y_1}, \cdots ,{y_k}\}$ be a set of $k \leqq n$ elements of ${F_n}$, where each ${y_i}$ is represented by a word ${Y_i}({x_1}, \cdots ,{x_n})$ in the generators ${x_j}$. Let $\partial {y_i}/\partial {x_j}$ denote the free derivative of ${y_i}$ with respect to ${x_j}$, and let ${J_{kn}} = ||\partial {y_i}/\partial {x_j}||$ denote the $k \times n$ Jacobian matrix. Theorem. If $k = n$, the set $\{ {y_1}, \cdots ,{y_n}\}$ generates ${F_n}$ if and only if ${J_{nn}}$ has a right inverse. If $k < n$, the set $\{ {y_1}, \cdots ,{y_k}\}$ may be extended to a set of elements which generate ${F_n}$ only if ${J_{kn}}$ has a right inverse. Several applications are given.


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Keywords: Free calculus, primitive elements, free basis, invertible matrices
Article copyright: © Copyright 1973 American Mathematical Society