# On a problem of Freudenthal's

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On a problem of Freudenthal's COMPOSITIO MATHEMATICA HERBERTABELS On a problem of Freudenthal’s Compositio Mathematica, tome 35, no 1 (1977), p. 39-47. <http://www.numdam.org/item?id=CM_1977__35_1_39_0> © Foundation Compositio Mathematica, 1977, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http:// http://www.compositio.nl/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commer- ciale ou impression systématique est constitutive d’une infraction pé- nale. Toute copie ou impression de ce fichier doit contenir la pré- sente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ 39 ON A PROBLEM OF FREUDENTHAL’S Herbert Abels COMPOSITIO MATHEMATICA, Vol. 35, Fasc. 1, 1977, pag. 39-47 Noordhofl International Publishing Printed in the Netherlands 1 Let G be a finitely generated group, let F(G) be the compact space of its ends. The group G acts on F(G). If the cardinality of F(G) is infinite Freudenthal [5] showed that there is at most one fixed point in F(G). He left open the following QUESTION: Are there finitely generated groups G with a fixed point in F(G) and F(G) infinite? The answer is no. This follows from Stallings’s (cf. [7]) structure theorem on finitely generated groups with infinitely many ends. Together with Freuden- thal’s result we have a more precise statement. A G-space X is called minimal if any closed G-stable subset of X is empty or equal to X, equivalently: if the orbit of any point x E X is dense in X. THEOREM 1: Let G be a finitely generated group. The space F(G) of ends of G is a minimal G-space, except when F(G) consists of two points which are fixed. One can ask the corresponding question for locally compact groups. Here the answer is yes. The structure of the occurring groups is completely described in terms of generators and relations (s. Theorem 2

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