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Proceedings of the American Mathematical Society
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
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}} = \vert\vert\partial {y_i}/\partial {x_j}\vert\vert$ 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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DOI: http://dx.doi.org/10.1090/S0002-9939-1973-0330295-8
PII: S 0002-9939(1973)0330295-8
Keywords: Free calculus, primitive elements, free basis, invertible matrices
Article copyright: © Copyright 1973 American Mathematical Society