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Geometric realizations for free quotients
Published online by Cambridge University Press: 09 April 2009
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In [7] Lyndon introduced the concept of inner rank for groups. He defined the inner rank of an arbitrary group G to be the upper bound of the ranks of free homomorphic images of G. Both Lyndon and Jaco have shown that the inner rank of the fundamental group of a closed 2-manifold with Euler characteristic 2 − p, p ≧ 0, is [p/2] where [p/2] is the greatest integer ≦ p/2. The proof given by lyndon [8] uses algebraic techniques; whereas, the proof by Jaco [4] is geometrical.
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- Research Article
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- Journal of the Australian Mathematical Society , Volume 14 , Issue 4 , December 1972 , pp. 411 - 418
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- Copyright © Australian Mathematical Society 1972
References
[4]Jaco, W., ‘Heegaard splitting and splitting homomorphisms’, Trans. A. M. S., Vol. 146 (1969), 365–375.CrossRefGoogle Scholar
[5]Jaco, W., ‘Non-retractible cubes-with-holes’, Michigan Math. J., Vol. 18 (1971), 193–201.CrossRefGoogle Scholar
[7]Lyndon, R. C., ‘The equation a2 b2 = c2 in free groups’, Mich. Math. J. 6 (1959), 89–95.CrossRefGoogle Scholar
[8]Lyndon, R. C., ‘Dependence in groups,’ Colloq, Mathe. (Warsaw) XIV (1966), 275–283.CrossRefGoogle Scholar
[9]Markov, A., ‘The insolubility of the problem of homeomorphy’, Dokl. Akad. Nauk SSSR 121 (1958), 218–220.Google Scholar
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