Wolfgang WOESS


Institut für Mathematische Strukturtheorie (Math C)
Technische Universität Graz
Steyrergasse 30
8010 Graz, Austria

phone: ++43 (0)316 873-7130
fax: ++43 (0)316 873-4507
email: woess[at]TUGraz[dot]at



                   W.W.

                 

Contents

- About myself
- Research interests
- FWF projects
- Books
- Papers since 1994 (downloadable)
- Conferences
- Coauthors
- Refereeing
- Teaching
- Politics
- F.Woess (1915-1995),
- E.Woess (1917-2001)


NEW:

The Austrian Science Fund (FWF) has approved the proposal for funding a graduate college (DK-plus) with the title "Discrete Mathematics". This is a joint effort of mathematicians at Technische Universität Graz, Karl Franzens Universität Graz and Montanuniversität Leoben. I am the speaker of this large project. The funding is substantial. There will be several PhD position openings, starting with October, 2010. See www.math.tugraz.at/discrete for more details.



Link (in German): Unterlagen der Curriculakommission für Doktoratsstudien und ULG der TU Graz (I'm currently head of the Committee for Doctoral Studies of TU Graz, relevant documents can be found at this link.)




This picture was taken at the conference
"Geometric Group Theory, Random Walks, and Harmonic Analysis"
in Cortona (Italy), 13-19 June 2004.
You can have a look at the picture gallery of that conference.


About myself

I was born in Vienna in 1954. I studied Mathematics at Vienna University of Technology (MSc - "Diplomingenieur") and the University of Salzburg (PhD).

For 6 years (1982-88) I was Assistant Professor ("Universitätsassistent") at the Institut für Mathematik und Angewandte Geometrie of the Montanuniversität Leoben in Austria.

In 1984/85, I was on leave for a year at the Dipartimento di Matematica "G. Castelnuovo" of the University of Rome_I.

A few years later, in 1988, I moved to Italy permanently. For 6 years (1988-94) I was associate professor at the Dipartimento di Matematica "F. Enriques" of the University of Milan, then for 4 years full professor at the same University and, in the last year, at the Dipartimento di Matematica e Applicazioni of the newly founded Universita` di Milano-Bicocca.

I married in 1988. Here are pictures of my wife Susanne (Sanni), our daughter Julia (born in June 1998) and myself, including a less formal picture of mine.

In September 1999 we came back to Austria. I am now a full professor in Institut für Mathematische Strukturtheorie (formerly Institut für Mathematik C) of Technische Universität Graz in Austria.

From 2000 to 2003, I served as the Secretary ("Schriftführer") of the Austrian Mathematical Society ÖMG.

My Erdös number is 2, thanks to Carsten Thomassen.

You can look at my mathematical genealogy.

You can also have a look at some pictures that I took during my 50th birthday hike in the Styrian alps, to see how nice Styria is. I also have a small photo gallery of Graz pictures for all those who might want to get an idea how the town looks like, and a photo gallery of pictures from the beautiful wine and castle hills South of Graz

On the last day of my trip to Japan in summer 2007, I prepared a small page on my Japanese travel experiences (including a few pictures).

My 55th birthday hike pictures are from the Tyrolean mountains.

REPORT ON THE ACTIVITIES OF MY RESEARCH GROUP IN THE PERIOD 1999-2006 (pdf)

Curriculum vitae and List of publications (pdf files).

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Research interests

- Random walks on infinite graphs and groups
- Discrete potential theory
- Boundary theory and harmonic functions
- Structure of infinite graphs, groups and formal languages

Here are two introductory papers (pdf) to the topic of random walks: One by myself, at a very elementary level and in German (from the book "Zur Kunst des formalen Denkens"), and the other, more advanced, by L. Saloff-Coste (from the Notices of the American Mathematical Society).

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Austrian Science Fund projects

From October 2002 to July 2006, I have run the project entitled "Asymptotic Properties of Random Walks on Graphs" which was funded by the FWF (Austrian Science Fund), project number FWF-P15577-N05. Funding consisted primarily of the salaries for PostDoc or PhD fellows.
You can have a look at the project description and the final report (pdf files). The FWF requires final reviews from the project referees. The - very friendly - project referee has written a very nice final review (pdf file). I only have vague ideas who this referee might be, but in any case would like to express my gratitude here.

Since October 2006, I am running a second FWF project entitled "Random walks, random configurations, and horocyclic products", project number FWF-P19115-N18. Here is the project description (pdf).

Besides this, there is the Doctoral College (DKplus) "Discrete Mathematics" starting in 2010, of which I am the speaker.

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Three books and three proceedings volumes

  1. W. Woess: Denumerable Markov Chains - Generating Functions, Boundary Theory, Random Walks on Trees , EMS Textbooks in Mathematics, European Mathematical Society Publishing House, xviii+351 pages, 2009.

  2. W. Woess: Random Walks on Infinite Graphs and Groups, Cambridge Tracts in Mathematics 138, Cambridge University Press, 334+xi pages, 2000. Paperback re-edition, 2008.
    ERRATA - Three pages of "errata" regarding various misprints and two "true" mistakes.
    Book reviews: Mathematical Reviews (pdf), Zentralblatt (ps), Bull. London Math. Soc. (pdf; attention - text starts on bottom of page); Acta Sci. Math. (Szeged) (ps), Bulletin Amer. Math. Soc. (pdf), Nieuw Archief voor Wiskunde (pdf).

  3. W. Woess: Catene di Markov e Teoria del Potenziale nel Discreto, Quaderni dell' Unione Matematica Italiana 41, 165+iii pages, 1996.
    Book review: Zentralblatt (ps).

  4. V. A. Kaimanovich with K. Schmidt and W. Woess (editors): Random Walks and Geometry, Proceedings (ESI, Vienna, 2001), de Gruyter, Berlin, 532+x pages, 2004.

  5. P. M. A. Grabner, W. Woess (editors): Fractals in Graz 2001: Analysis - Dynamics - Geometry - Stochastics, Proceedings (Graz 2001), Birkhäuser, Basel, 283+iii pages, 2003.

  6. M. A. Picardello, W. Woess (editors): Random Walks and Discrete Potential Theory, Proceedings (Cortona 1997), Symposia Mathematica XXXIX, Cambridge University Press, 361+ix pages, 1999.

    7. Back to table of contents.

Papers since 1994, plus miscellaneous older ones (downloadable)

  1. W. Huss, E. Sava and W. Woess: Entropy sensitivity of languages defined by infinite automata, via Markov chains with forbidden transitions, preprint (2009). (pdf)

  2. W. Woess: Context-free pairs of groups. II - Cuts, tree sets, and random walks, preprint (2009). (pdf)

  3. T. Ceccherini-Silberstein and W. Woess: Context-free pairs of groups. I - Context-free pairs and graphs, preprint (2009). (pdf)

  4. A. Bendikov, L. Saloff-Coste, M. Salvatori and W. Woess: The heat semigroup and Brownian motion on strip complexes, preprint (2008/09). (pdf)

  5. F. Lehner, M. Neuhauser and W. Woess: On the spectrum of lamplighter groups and percolation clusters, Mathematische Annalen 342 (2008) 69-89. (pdf)

  6. L. Saloff-Coste and W. Woess: Computations of spectral radii on G-spaces, Contemporary Math. 484 (2009) 195-218. (pdf)

  7. M. Peigné and W. Woess: On recurrence of reflected random walk on the half-line, with an appendix on results of Martin Benda, preprint, TU Graz (2006). (pdf) [This paper needs a thorough revision.]

  8. L. Bartholdi, M. Neuhauser and W. Woess: Horocyclic products of trees, J. European Math. Society 10 (2008) 771-816. (pdf)

  9. D. I. Cartwright and W. Woess: The spectrum of the averaging operator on a network (metric graph), Illinois J. Math. 51 (2007) 805-830. (pdf)

  10. A. Karlsson and W. Woess: The Poisson boundary of lamplighter random walks on trees, Geometriae Dedicata 124 (2007) 95-107. (pdf)

  11. R. Ortner and W. Woess: Non-backtracking random walks and cogrowth of graphs, Canadian J. Math.59 (2007) 828-844. (pdf)

  12. S. Brofferio and W. Woess: Positive harmonic functions for semi-isotropic random walks on trees, lamplighter groups, and DL-graphs, Potential Analysis 24 (2006) 245-265. (pdf)

  13. L. Saloff-Coste and W. Woess: Transition operators on co-compact G-spaces, Revista Matematica Iberoamericana 22 (2006) 747-799. (pdf)

  14. S. Brofferio and W. Woess: Green kernel estimates and the full Martin boundary for random walks on lamplighter groups and Diestel-Leader graphs, Annales Inst. H. Poincaré (Prob. & Stat.) 41 (2005) 1101-1123 (pdf), Erratum (pdf).

  15. W. Woess: A note on the norms of transition operators on lamplighter graphs and groups, Int. J. Algebra and Computation 15 (2005) 1261-1272. (pdf)

  16. L. Bartholdi and W. Woess: Spectral computations on lamplighter groups and Diestel-Leader graphs, J. Fourier Analysis Appl. 11 (2005) 175 - 202. (pdf)

  17. W. Woess: Lamplighters, Diestel-Leader graphs, random walks, and harmonic functions, Combinatorics, Probability & Computing 14 (2005) 415-433. (ps)

  18. D. I. Cartwright and W. Woess: Isotropic random walks in a building of type Ad~, Math. Zeitschrift 247 (2004) 101-135. (pdf)

  19. W. Woess: Generating function techniques for random walks on graphs, in "Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces", P. Auscher, Th. Coulhon and A. Grigor'yan, eds. Contemporary Math. 338 (2003) 391-423. (ps)

  20. T. Ceccherini-Silberstein and W. Woess: Growth-sensitivity of context-free languages, Theoretical Comp. Sci. 307 (2003), 103-116. (pdf)

  21. T. Ceccherini-Silberstein and W. Woess: Growth and ergodicity of context-free languages, Transactions Amer. Math. Soc. 354 (2002) 4597-4625. (ps)

  22. T. Nagnibeda and W. Woess: Random walks on trees with finitely many cone types, J. Theoret. Probab. 15 (2002) 399-438. (pdf)

  23. V. A. Kaimanovich and W. Woess: Boundary and entropy of space homogeneous Markov chains, Ann. Probab. 30 (2002) 323-363. (pdf)

  24. W. Woess: Heat diffusion on homogeneous trees, Bollettino Un. Mat. It. 4-B (2001) 703-709 and Erratum (caused by printer), Bollettino Un. Mat. It. 5-B (2002) 259-260. (ps)

  25. S. Brofferio and W. Woess: On transience of card shuffling, Proc. Amer. Math. Soc. 129 (2001) 1513-1519. (ps)

  26. N. Seifter and W. Woess: Approximating graphs with polynomial growth, Glasgow Math. J. 42 (2000) 1-8. (pdf)

  27. W. Woess: A note on tilings and strong isoperimetric inequality, Math. Proc. Cambridge Phil. Soc. 124 (1998) 385-393. (pdf)

  28. L. Saloff-Coste and W. Woess: Transition operators, groups, norms, and spectral radii, Pacific J. Math. 180 (1997) 333-367. (ps)

  29. P. J. Grabner and W. Woess: Functional iterations and periodic oscillations for random walk on the Sierpinski graph, Stochastic Proc. Appl. 69 (1997) 127-138. (pdf)

  30. L. Saloff-Coste and W. Woess: Computing norms of group-invariant transition operators, Combinatorics, Probability, and Computing 5 (1996) 161-178. (ps)

  31. W. Woess: Dirichlet problem at infinity for harmonic functions on graphs, International Conference on Potential Theory 1994, Proceedings (editors: J. Kral et al.), de Gruyter, Berlin (1996) 189-217. (ps)

  32. W. Woess: The Martin boundary for harmonic functions on groups of automorphisms of a homogeneous tree, Monatshefte für Mathematik 120 (1995) 55-72. (pdf)

  33. W. Woess: Topological groups and recurrence of quasi transitive graphs, Rendiconti Sem. Mat. Fis. Milano (1994) 185-213. (pdf)

  34. M. A. Picardello and W. Woess: The full Martin boundary of the bi-tree, Ann. Probab. 22 (1994) 2203-2222. (pdf)

  35. D. I. Cartwright, V. A. Kaimanovich and W. Woess: Random walks on the affine group of local fields and of homogeneous trees, Ann. Institut Fourier (Grenoble) 44 (1994) 1243-1288. (pdf)


    36. Here are miscellanous older papers in electronic form:


  36. C. Thomassen and W. Woess: Vertex-transitive graphs and accessibility, J. Combinatorial Th., Ser. B. 58 (1993) 248-268. (pdf)

  37. D. I. Cartwright, P. M. Soardi and W. Woess: Martin and end compactifications for nonlocally finite graphs, Transactions Amer. Math. Soc. 338 (1993) 679-693. (pdf)

  38. S. Giulini and W. Woess: The Martin compactification of the Cartesian product of two hyperbolic spaces, J. reine angew. Mathematik 444 (1993) 17-28. (pdf)

  39. W. Woess: Fixed sets and free subgroups of groups acting on metric spaces, Math. Zeitschrift 214 (1993) 425-440. (pdf)

  40. M. A. Picardello and W. Woess: Martin boundaries of Cartesian products of Markov chains, Nagoya Math. J. 128 (1992) 153-169. (pdf)

  41. D. I. Cartwright and W. Woess: Infinite graphs with nonconstant Dirichlet finite harmonic functions, SIAM J. Discrete Math. 5 (1992) 380-385. (pdf)

  42. P. M. Soardi and W. Woess: Amenability, unimodularity, and the spectral radius of random walks on infinite graphs, Math. Zeitschrift 205 (1990) 471-486. (pdf)

  43. M. A. Picardello and W. Woess: A converse to the mean value property on homogenous trees, Transactions Amer. Math. Soc. 311 (1989) 209-225. (pdf)

  44. W. Woess: Graphs and groups with tree-like properties, J. Combinatorial Th., Ser. B, 47 (1989) 361-371. (pdf)

  45. M. A. Picardello and W. Woess: Martin boundaries of random walks: ends of trees and groups, Transactions Amer. Math. Soc. 302 (1987) 185-205. (pdf)

  46. W. Woess: Nearest neighbour random walks on free products of discrete groups, Bollettino Unione Mat. Italiana 5-B (1986) 691-982. (pdf, relatively large scan file)

  47. W. Woess: A short computation of the norms of free convolution operators, Proc. Amer. Math. Soc. 96 (1986) 167-170. (pdf)

  48. M. A. Picardello and W. Woess: Random walks on amalgams, Monatshefte für Mathematik 100 (1985) 21-33. (pdf)

  49. W. Woess: Random walks and periodic continued fractions, Advances in Applied Probability 17 (1985) 67-84. (pdf, relatively large scan file)

  50. W. Woess: Aperiodische Wahrscheinlickeitsmaße auf topologischen Gruppen, Monatshefte für Mathematik 90 (1980) 339-345. (pdf)


Here you can find my full list of publications, including a few comments (mainly concerning specific experiences that I had with some papers).

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Conferences

In the last years, I have been the (co-)organizer of several conferences about random walks and related subjects.

Random Walks and Discrete Potential Theory

This conference took place in the week June 22 - June 28, 1997. It was organized by Vadim A. Kaimanovich, Massimo Picardello, Laurent Saloff-Coste and myself. You can have a look at the ps-file containing program and list of participants. The proceedings volume Random Walks and Discrete Potential Theory was edited by Massimo Picardello and myself.

2001 - Random Walks in Vienna

In the period February - July 2001, Vadim Kaimanovich (Rennes), Klaus Schmidt (Vienna) and myself have organized special semester on RANDOM WALKS at the Erwin Schrödinger Institute in Vienna. Deatils of the program can be found here.

Fractals in Graz 2001

In the week June 4--9, 2001, there was a conference with this title (subtitle: Analysis - Dynamics - Geometry - Stochastics) at the Technical University of Graz, as a complement to the ESI programme on Random Walks. The organizing committee consisted of Martin Barlow (Vancouver), Robert Strichartz (Ithaca), Peter Grabner (Graz) and myself. For further informations, see http://finanz.math.tu-graz.ac.at/~fractal/.

Cortona conference in 2004

In the week 13-19 June 2004, there was a conference on "Geometric Group Theory, Random Walks, and Harmonic Analysis" in Cortona (Italy). For more information (pictures !), click on the conference title.

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Special session in Mainz, 2005

In the days 16-19 June 2005, there was a Joint meeting of AMS, DMV and ÖMG at Mainz, Germany. (ÖMG=Austrian Math. Soc.) Within that meeting, a joint session on Stochastic Analysis on Metric Spaces was organized by Laurent Saloff-Coste (Cornell Univ.), Karl-Theodor Sturm (Univ. Bonn) and myself. Main speakers at the special session were B. Driver (UC San Diego), F. Götze (Univ. Bielefeld), G. Lawler (Cornell Univ.) and K.-Th. Sturm (Univ. Bonn).

ESI workshop in spring 2006

In the two weeks March 12-25 March 2006, there was an RDSES/ESI Educational Workshop on Discrete Probability at the Erwin Schrödinger Institute (ESI) in Vienna. Organzers: V.A. Kaimanovich (Bremen), K. Schmidt (Vienna) and myself. Among other, there will be 4 minicourses, to be held by P. Diaconis, A. Gamburd, G. Lawler, and Ch. Pittet.

Workshop on "boundaries"
June 29 - July 4, 2009, TU Graz

within the EU Curie program "GROUPS: European training courses and conferences in group theory" (but the EU money did not arrive so far)

Conference webpage

Scientific committee: Tatiana Nagnibeda, Christophe Pittet, Hamish Short, Wolfgang Woess

Local committee: Ecaterina Sava, Wilfried Huss, Wolfgang Woess

List of selected speakers and participants: Alexander BENDIKOV, Philippe BOUGEROL, Emmanuel BREUILLARD, Donald CARTWRIGHT, Manfred DENKER, Ewa DAMEK, Martin DUNWOODY, Anna ERSCHLER, Vadim KAIMANOVICH, Francois LEDRAPPIER, Volodimyr NEKRASHEVICH, Panos PAPAZOGLOU, James PARKINSON, Massimo PICARDELLO, Tim RILEY, Klaus SCHMIDT, Tim STEGER, Anton THALMAIER, Anatoly VERSHIK

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Coauthors

Laurent Bartholdi
Sara Brofferio
Alexander Bendikov
Tullio Ceccherini-Silberstein
Donald I. Cartwright
Franz Lehner
Peter Gerl
Saverio Giulini
Peter J. Grabner
Chris D. Godsil
Wilfried Imrich
Vadim A. Kaimanovich
Anders Karlsson
Franz Lehner
Bojan Mohar
Tatiana Nagnibeda
Markus Neuhauser
Ronald Ortner
Marc Peigné
Massimo A. Picardello
Laurent Saloff-Coste
Maura Salvatori
Norbert Sauer
Norbert Seifter
Paolo M. Soardi
Mitch Taibleson
Carsten Thomassen
Mark E. Watkins

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Refereeing

Throughout the years, I have acted as the referee for approximately as many papers as I have published myself, for the following journals (resp. conferences):

Abhandlungen des Mathematischen Seminars der Universität Hamburg;
Advances in Applied Mathematics;
Annales Blaise Pascal;
Annales de l'Institut Fourier (Grenoble);
Annales de l'Institut Poincaré-B;
Annali di Matematica Pura ed Applicata;
Annals of Probability;
Archiv der Mathematik;
Bollettino dell'Unione Matematica Italiana;
Bulletin de Sciences Math\'ematiques;
Bulletin of the London Mathematical Society;
Circuits, Systems, and Signal Processing;
Combinatorics, Probability, and Computing;
Computing;
Contemporary Mathematics;
Discrete and Computational Geometry;
Discrete Mathematics;
Discussiones Mathematicae;
Electronic Comunications in Probability;
Electronic Journal of Probability;
Glasgow Mathematical Journal;
ICALP;
Illinois Journal of Mathematics;
International Journal of Algebra and Computation;
International Journal of Mathematics and Mathematical Sciences;
International Mathematical Research Notes;
Israel Journal of Mathematics;
Journal of Algebraic Combinatorics;
Journal of Combinatorial Theory-B;
Journal of Functional Analysis;
Journal of Graph Theory;
Journal of Group Theory;
Journal of Lie Theory;
Journal of Mathematical Analysis and Applications;
Journal of Statistical Physics;
Journal of the American Mathematical Society.
Journal of the London Mathematical Society;
Journal of Theoretical Probability;
Linear and Multilinear Algebra;
Mathematische Annalen;
Mathematische Nachrichten;
Mathe\-matische Zeitschrift;
Monatshefte für Mathematik;
Nagoya Mathematical Journal;
Pacific Journal of Mathematics;
Potential Analysis;
Probability Theory and Related Fields;
Proceedings of the London Mathematical Society;
Random Structures and Algorithms;
STACS;
Transactions of the American Mathematical Society.

From 1990 to 1999 I have served as an Associate Editor of the journal ``Circuits, Systems and Signal Processing'' (Birkhäuser).

In 2007, I have acted as a referee in the evaluation of the Mathematical Insitute of the Slovak Academy of Sciences.

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Teaching

Please see the relative informations on TUG online or the item Lehre / Teaching on our own homepage.

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On politics

In February 2000, Austria's new government, a right wing coalition between Christian Democrats and the (so-called) Freedom Party had led the country into a tempest of national and international criticism and worldwide isolation. This had a particular impact on the scientific comunity, whose work is based on international exchange and cultural openmindendness. This is why, after those events, I had included my personal considerations on Austrian politics in my scientific homepage.

In the meantime, Austria has disappeared from most international news, and there have been only few news flashes reporting that after only two and a half years, and a full year before the end of its term, the government had collapsed in autumn 2002 because of a major crisis within the Freedom Party. Since 2003, the same coalition was back to power again, but with the Freedom Party down to 10% instead of the previous 27%. Since 2006, after the Freedom Party has disassembled in two pieces, they are no more represented in the government, and we are now back to the "big coalition" between Social and Christian Democrats, even though these two parties appear to hate each other profoundly.

The original text (in particular the one regarding the so-called "sanctions") is now really out of date, so I removed it at last.   Who is interested may contact me by email, and I will send it. As a matter of fact, while probably most Austrian colleagues considered this a peculiar thing to put on my webpage, I had quite a bit of positive feedback from abroad.

Regarding Italian politics (having lived in Italy from 1988 until 1999), I believe that Mr. Berlusconi has been far more dangerous for democracy than Haider, having corrupted, if not destroyed Italy's civil society. Maybe the day will come when he is captured during a clandestine attempt to cross the border from Italy to Switzerland close to lake Como. (Decipher this hint at a historical event...)
On April 5, 2006, a few days before the elections in Italy, I was really shocked that the Christian democrat Austrian prime minister in a meeting of European popular and christian democratic parties had wished success to Mr. Berlusconi, at the same time when Europe was taking (timid, but still) measures against Mr. Lukashenko, the Bielorussian analogue of Berlusconi.
In the meantime (May 2008) I have to face the sad truth that Italy is back to a stable regime of new fascism. The brainwashing that Berlusconi could force on the people via his undemocratical TV power has had its lasting effects, and my hope regarding the repetition of a historic event close to Lake Como has faded away. All this is particularly sad, if I remember the optimistic atmosphere of the "mani pulite" years when one could believe in the birth of a renewed civil society in Italy.

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Friedrich Woess (1915-1995)
Elisabeth Woess (1917-2001)

My father, Friedrich Woess, was Professor at Universität für Bodenkultur, Vienna.
He was also gifted in painting (mostly water color). He left about 850 pictures, 99% are landscapes.
I have organized a show of some of them in Summer, 2002.
There was another show in Summer, 2006 (originally planned for 2005).
You can have a look at some of his paintings: those from the exhibition in 2006, and others: part1, part2, part3, part4, all four from two booklets with reproductions from the 1990ies, and Greece 1979.
You can read an accompanying text (in German), and find the announcement of the abovementioned first exhibition in 2002.

My mother, Elisabeth Woess (1917-2001) was also a university professor, at the Institute of Botanics of Vienna University. You can see a few late photographs of her and read an obituary (pdf, in German) that appeared in Verh. Zool.-Bot. Ges. Österreich 138 (2001) 275-278. A posthumuous publication has appeared in Plant Systematics and Evolution 258 (2006) 27-37. It has a small background story: my mother had continued her research work until a week before she passed a way at the age of 84. This was supposed to become a joint work with a Professor in Florida of Hungarian origin who years before had sent her specimen of an antarctic lichens to be studied and classified. In the last year of her life she often expressed sadness about the fact that she had finished her part of this work, but her colleague would not complete it. (Indeed, he had retired, too, and had no more laboratory access.) I found the manuscript and drawings in the material that she left behind, and managed to convince the director of the Botanical Insitute in Vienna to take care of its publication. It was carefully revised and re-edited. See in particular the Comments and Acknowledgements at the end of the paper.

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Last modified on January 26, 2010.

Impressum:
Für den Inhalt verantwortlich: Wolfgang Woess
Institut für Mathematische Strukturtheorie (Math C)
TU Graz, Steyrergasse 30, A-8010 Graz