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Structure Theory of Totally Disconnected Locally Compact Groups via Graphs and Permutations

Published online by Cambridge University Press:  20 November 2018

Rögnvaldur G. Möller
Rögnvaldur G. Möller*
Affiliation:
Science Institute, University of Iceland, IS-107 Reykjavik, Iceland, e-mail: roggi@raunvis.hi.is
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Abstract

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Willis's structure theory of totally disconnected locally compact groups is investigated in the context of permutation actions. This leads to new interpretations of the basic concepts in the theory and also to new proofs of the fundamental theorems and to several new results. The treatment of Willis's theory is self-contained and full proofs are given of all the fundamental results.

Keywords

22D0520B0720B2705C25totally disconnected locally compact groupsscale functionpermutation groupsgroups acting on graphs
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 54 , Issue 4 , 01 August 2002 , pp. 795 - 827
Copyright
Copyright © Canadian Mathematical Society 2002

References

[1] Bergman, G. M. and Lenstra, H. W., Subgroups dose to normal subgroups. J. Algebra 127(1989), 80–97.Google Scholar
[2] Bhattacharjee, M. and Macpherson, D., Strange permutation representations offreegroups. J. Austral. Math. SOG, to appear.Google Scholar
[3] Bhattacharjee, M., Macpherson, D., Möller, R. G. and Neumann, P. M., Notes on Infinite Permutation Croups. Hindustan Book Agency, Delhi, India, 1997, Republished as Springer Lecture Notes in Math., 1698, Springer, 1998.Google Scholar
[4] Bourbaki, N., Elements of mathematics: general topology. Palo Alto, London, 1966, English translation of Elements de mathematique, topologie generale, Paris, 1942.Google Scholar
[5] Cameron, P. J., Praeger, C. E. and Wormald, N. C., Infinite highly arc transitive digraphs and universal coveringdigraphs. Combinatorica 13(1993), 377–396.Google Scholar
[6] van Danteig, D., Zur topologischen Algebra III. Brouwersche und Cantorsche Gruppen. Compositio Math. 3(1936), 408-26.Google Scholar
[7] Dixon, J. D. and Mortimer, B., Permutation Croups. Graduate Texts in Math. 163, Springer 1996.Google Scholar
[8] Glöckner, H., Scale functions on linear groups over local skewfields. J. Algebra 205(1998), 525–541.Google Scholar
[9] Glöckner, H., Scale functions on p-adic He groups. Manuscripta Math. 97(1998), 205–215.Google Scholar
[10] Hewitt, E. and Ross, K. A., Abstract Harmonie analysis, Volume I. Springer, Berlin-Göttingen-Heidelberg 1963. [II] P. J. Higgins, An Introduction to Topological Groups, Cambridge University Press, Cambridge, 1974.Google Scholar
[12] Jaworski, W., Rosenblatt, J. and Willis, G., Concentration functions on locally compact groups. Math. Ann. 305(1996), 673–691.Google Scholar
[13] Kepert, A. and Willis, G., Scale funetion and tree ends. J. Aust. Math. Soc. 70(2001), 273–292.Google Scholar
[14] Möller, R. G., Descendants in highly arc transitive digraphs. Discrete Math. 247(2002), 147–157.Google Scholar
[15] Nebbia, C., Minimally almostperiodic totally disconnected groups. Proc. Amer. Math. Soc. 128(2000), 347–351.Google Scholar
[16] Ross, K. A. and Willis, G., Riemann sums and modular functions on locally compact groups. Pacific J.Math. 180(1997), 325–331.Google Scholar
[17] Schlichting, G., Polynomidentitäten und Permutationsdarstellungen lokalkompacter Gruppen. Invent. Math. 55(1979), 97–106.Google Scholar
[18] Schlichting, G., Operationen mit periodischen Stabilisatoren. Arch. Math. 34(1980), 97–99.Google Scholar
[19] Trofimov, V. I., Automorphism groups ofgraphs as topological groups. Math. Notes 38(1985), 717–720.Google Scholar
[20] Willis, G., The strueture of totally disconnected, locally compact groups. Math. Ann. 300(1994), 341–363.Google Scholar
[21] Willis, G., Totally disconnected groups andproofs of conjeetures of Hofmann and Mukherjea. Bull. Austral. Math. Soc. 51(1995), 489-194.Google Scholar
[22] Willis, G., Further properties ofthe scale function on a totally disconnected group. J. Algebra 237(2001), 142–164.Google Scholar
[23] Woess, W., Topological groups and infinite graphs. In: Directions in Infinite Graph Theory and Combinatorics, (ed. R. Diestel), Topics in Discrete Math. 3, North Holland, Amsterdam 1992, also in Discrete Math. 95(1991), 373–384.Google Scholar