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Scale functions and tree ends

Part of:Locally compact groups and their algebras

Published online by Cambridge University Press:  09 April 2009

A. Kepert and
G. Willis
A. Kepert
Affiliation:
School of Science and Technology University of NewcastleOurimbah NSW 2258Australia e-mail: andrew@maths.newcastle.edu.au
G. Willis
Affiliation:
School of Mathematical and Physical Sciences University of NewcastleCallaghan NSW 2308Australia e-mail: george@maths.newcastle.edu.au
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Abstract

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A class of totally disconnected groups consisting of partial direct products on an index set is examined. For such a group, the scale function is found, and for automorphisms arising from permutations of the index set, the tidy subgroups are characterised. When applied to the case where the index set is a finitely-generated free group and the permutation is translation by an element x of the group, the scale depends on the cyclically reduced form of x and the tidy subgroup on the element which conjugates x to its cyclically reduced form.

MSC classification

Secondary:22D05: General properties and structure of locally compact groups

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 70 , Issue 2 , April 2001, pp. 273 - 292
Copyright
Copyright © Australian Mathematical Society 2001

References

[1]Bergman, G. M., ‘On groups acting on locally finite graphs’, Ann. of Math. 88 (1968), 335–340.CrossRefGoogle Scholar
[2]Bhattacharjee, M. and Macpherson, D., ‘Strange permutation representations of free groups’, preprint (1999).Google Scholar
[3]Dicks, W. and Dunwoody, M. J., Groups acting on graphs (Cambridge University Press, Cambridge, 1989).Google Scholar
[4]Freudenthal, H., ‘Uber die enden diskreter Räume and Gruppen’, Comment. Math. Helv. 17 (1944), 1–38.CrossRefGoogle Scholar
[5]Glöckner, H., ‘Scale functions on p-adic Lie groups’, Manuscripta Math. 97 (1998), 205–215.Google Scholar
[6]Glöckner, H. and Willis, G. A., ‘Uniscalar p-adic Lie groups’, Forum Math., to appear.Google Scholar
[7]Hofmann, K. H., Liukkonen, J. R. and Mislove, M. W., ‘Compact extensions of compactly generated nilpotent groups are pro-Lie’, Proc. Amer. Math. Soc. 84 (1982), 443–448.CrossRefGoogle Scholar
[8]Hopf, H., ‘Enden offener Räume und unendliche diskontinuierliche Gruppen’, Comment. Math. Helv. 16 (1943), 81–100.CrossRefGoogle Scholar
[9]Magnus, W., Karrass, A. and Solitar, D., Combinatorial group theory (Dover, New York, 1975).Google Scholar
[10]Palmer, T., Banach algebras and a general theory of *-algebras, vols I, II (Cambridge University Press, Cambridge, 1994, 2001).CrossRefGoogle Scholar
[11]Parreau, A., ‘Sous-groupes elliptiques de groupes linéaires sur un corps valué’, preprint (1999).Google Scholar
[12]Stallings, J. R., ‘On torsion-free groups with infinitely many ends’, Ann. of Math. 88 (1968), 312–334.CrossRefGoogle Scholar
[13]van Dantzig, D., ‘Zur topologischen Algebra III. Brouwersche und Cantorsche Gruppen’, Compositio Math. 3 (1936), 408–426.Google Scholar
[14]Willis, G. A., ‘Further properties of the scale function on a totally disconnected group’, J. Algebra, to appear.Google Scholar
[15]Willis, G. A., ‘The structure of totally disconnected, locally compact groups’, Math. Ann. 300 (1994), 341–363.CrossRefGoogle Scholar
[16]Willis, G. A., ‘Totally disconnected nilpotent locally compact groups’, Bull. Austral. Math. Soc. 55 (1997), 143–146.CrossRefGoogle Scholar
[17]Woess, W., ‘Amenable group actions on infinite graphs’, Math. Ann. 284 (1989), 251–265.CrossRefGoogle Scholar