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Publisher:
Cambridge University Press
Online publication date:
November 2012
Print publication year:
2012
Online ISBN:
9781139176064

Book description

The theory of R-trees is a well-established and important area of geometric group theory and in this book the authors introduce a construction that provides a new perspective on group actions on R-trees. They construct a group RF(G), equipped with an action on an R-tree, whose elements are certain functions from a compact real interval to the group G. They also study the structure of RF(G), including a detailed description of centralizers of elements and an investigation of its subgroups and quotients. Any group acting freely on an R-tree embeds in RF(G) for some choice of G. Much remains to be done to understand RF(G), and the extensive list of open problems included in an appendix could potentially lead to new methods for investigating group actions on R-trees, particularly free actions. This book will interest all geometric group theorists and model theorists whose research involves R-trees.

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Contents

References
References
[1] R. C., Alperin and K. N., Moss, Complete trees for groups with a real-valued length function, J. London Math. Soc. 31 (1985 Google Scholar), 55–68.
[2] R., Baer, The subgroup of the elements of finite order of an abelian group, Ann. Math. 37 (1936 Google Scholar), 766–781.
[3] S. A., Basarab, On a problem raised by Alperin and Bass. In: Arboreal Group Theory (ed. R. C., Alperin), MSRI Publications vol. 19, pp. 35–68, Springer-Verlag, New York, 1991 Google Scholar.
[4] H., Bass, Group actions on non-archimedean trees. In: Arboreal Group Theory (ed. R. C., Alperin), MSRI Publications vol. 19, pp. 69–131, Springer-Verlag, New York, 1991 Google Scholar.
[5] V. N., Berestovskii and C. P., Plaut, Covering ℝ-trees, ℝ-free groups, and dendrites, Adv. Math. 224 (2010 Google Scholar), 1765–1783.
[6] M., Bestvina and M., Feighn, Stable actions of groups on real trees, Invent. Math. 121 (1995 Google Scholar), 287–321.
[7] G., Cantor, Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen, J. Reine Angew. Math. 77 (1874), 258–262. Also in: Georg Cantor, Gesammelte Abhandlungen (ed. E., Zermelo), Julius Springer, Berlin, 1932 Google Scholar.
[8] G., Cantor, Über unendliche lineare Punktmannigfaltigkeiten V, Math. Ann. 21 (1883 Google Scholar), 545–591.
[9] I. M., Chiswell, Harrison's theorem for Λ-trees, Quart. J. Math. Oxford (2) 45 (1994 Google Scholar), 1–12.
[10] I. M., Chiswell, Introduction to Λ-Trees, World Scientific, Singapore, 2001 Google Scholar.
[11] I. M., Chiswell, A-free groups and tree-free groups. In: Groups, Languages, Algorithms (ed. A.V., Borovik), Contemp. Math. vol. 378, pp. 79–86, Providence (RI), Amer. Math. Soc., 2005 Google Scholar.
[12] I. M., Chiswell and T.W., Müller, Embedding theorems for tree-free groups, Math. Proc. Cambridge Philos. Soc. 149 (2010 Google Scholar), 127–146.
[13] I. M., Chiswell, T.W., Müller, and J.-C., Schlage-Puchta, Compactness and local compactness for ℝ-trees, Arch. Math. 91 (2008 Google Scholar), 372–378.
[14] D. E., Cohen, Combinatorial Group Theory: A Topological Approach, London Mathematical Society Student Texts vol. 14, Cambridge University Press, 1989 Google Scholar.
[15] P. M., Cohn, Algebra (Second Edition), Volume 3, John Wiley & Sons, Chichester, 1991 Google Scholar.
[16] M., Coornaert, T., Delzant, and A., Papadopoulos, Géeométrie et Théorie des Groupes, Lecture Notes in Mathematics vol. 1441, Springer, Berlin, 1990 Google Scholar.
[17] M. J., Dunwoody, Groups acting on protrees, J. London Math. Soc. 56 (1997 Google Scholar), 125–136.
[18] L., Fuchs, Abelian Groups, Pergamon Press, Oxford, 1960 Google Scholar.
[19] D., Gaboriau, G., Levitt, and F., Paulin, Pseudogroups of isometries of ℝ and Rips' Theorem on free actions on ℝ-trees, Israel J. Math. 87 (1994 Google Scholar), 403–428.
[20] E., Ghys and P., de la Harpe, Sur les Groupes Hyperboliques d'après Mikhael Gromov, Birkhäuser, Boston, 1990 Google Scholar.
[21] D., Gildenhuys, O., Kharlampovich, and A. G., Myasnikov, CSA groups and separated free constructions, Bull. Austral. Math. Soc. 52 (1995 Google Scholar), 63–84.
[22] L., Greenberg, Discrete groups of motions, Can. J. Math. 12 (1960 Google Scholar), 414–425.
[23] M., Gromov, Hyperbolic groups. In: Essays in Group Theory (ed. S. M., Gersten), Mathematical Sciences Research Institute Publications vol. 8, pp. 75–263, Springer-Verlag, New York, 1987 Google Scholar.
[24] N., Harrison, Real length functions in groups, Trans. Amer. Math. Soc. 174 (1972 Google Scholar), 77–106.
[25] P. J., Higgins, Notes on Categories and Groupoids, Van Nostrand Reinhold, London-New York-Melbourne, 1971. (Reprinted with a new preface by the author: Repr. Theory Appl. Categ. 7 Google Scholar (2005), 1–178 (electronic).)
[26] W., Imrich, On metric properties of tree-like spaces. In: Beiträge zur Graphentheorie und deren Anwendungen (ed. Sektion MARÖK der Technischen Hochschule Ilmenau), pp. 129–156, Oberhof, 1977 Google Scholar.
[27] A., Kertész, Einführung in die Transfinite Algebra, Birkhäuser Verlag, Basel–Stuttgart, 1975 Google Scholar.
[28] F., Levi, Arithmetische Gesetze im Gebiete diskreter Gruppen, Rend. Palermo 35 (1913 Google Scholar), 225–236.
[29] G., Levitt, Constructing free actions on ℝ-trees, Duke Math. J. 69 (1993 Google Scholar), 615–633.
[30] R. C., Lyndon and P. E., Schupp, Combinatorial Group Theory, Springer-Verlag, Berlin–Heidelberg, 1977 Google Scholar.
[31] W., Magnus, A., Karrass, and D., Solitar, Combinatorial Group Theory. Presentations of Groups in Terms of Generators and Relations, reprint of the 1976 second edition, Dover, Mineola, NY, 2004 Google Scholar.
[32] G. A., Margulis, Discrete Subgroups of Semisimple Lie Groups, Springer-Verlag, Berlin–Heidelberg, 1991 Google Scholar.
[33] J. C., Mayer, J., Nikiel, and L. G., Oversteegen, Universal spaces for ℝ-trees, Trans. Amer. Math. Soc. 334 (1992 Google Scholar), 411–432.
[34] J.W., Morgan and P. B., Shalen, Valuations, trees and degenerations of hyperbolic structures: I, Ann. of Math. (2) 122 (1985 Google Scholar), 398–476.
[35] J.W., Morgan and P. B., Shalen, Free actions of surface groups on ℝ-trees, Topology 30 (1991 Google Scholar), 143–154.
[36] T.W., Müller, A hyperbolicity criterion for subgroups of RJ(G), Abh. Math. Semin. Univ. Hambg. 80 (2010 Google Scholar), 193–205.
[37] T.W., Müller Google Scholar, Some contributions to the theory of RJ-groups. In preparation.
[38] T.W., Müller and J.-C., Schlage-Puchta, On a new construction in group theory, Abh. Math. Semin. Univ. Hambg. 79 (2009 Google Scholar), 193–227.
[39] A. G., Myasnikov and V. N., Remeslennikov, Exponential groups, II: extensions of centralizers and tensor completion of CSA-groups, Internat. J. Algebra Comput. 6 (1996 Google Scholar), 687–711.
[40] A. G., Myasnikov, V. N., Remeslennikov, and D., Serbin, Regular free length functions on Lyndon's free ℤ[t]-group Fℤ[t]. In: Groups, Languages, Algorithms (ed. A. V., Borovik), Contemp. Math. vol. 378, pp. 33–77, Providence (RI), Amer. Math. Soc., 2005 Google Scholar.
[41] M. H. A., Newman, On theories with a combinatorial definition of “equivalence“, Ann. of Math. (2) 43 (1942 Google Scholar) 223–243.
[42] F. S., Rimlinger, ℝ-trees and normalisation of pseudogroups, Exper. Math. 1 (1992 Google Scholar), 95–114.
[43] H. L., Royden, Real Analysis, Macmillan, New York, 1963 Google Scholar.
[44] H., Schubert, Categories, Springer-Verlag, Berlin–Heidelberg, 1972 Google Scholar.
[45] J.-P., Serre, Trees, Springer-Verlag, Berlin–Heidelberg, 1980 Google Scholar.
[46] H., Short, Notes on word hyperbolic groups. In: Group Theory from a Geometrical Viewpoint (eds. E., Ghys, A., Haefliger, and A., Verjovsky), World Scientific, Singapore, 1991 Google Scholar.
[47] W., Sierpiński, Cardinal And Ordinal Numbers, Monographs of the Polish Academy of Science vol. 34, Warsaw, 1958 Google Scholar.
[48] H. J. S., Smith, On the integration of discontinuous functions, Proc. London Math. Soc. 6 (1875 Google Scholar), 140–153.
[49] T., Szele, Ein Analogon der Körpertheorie für abelsche Gruppen, J. Reine Angew. Math. 188 (1950 Google Scholar), 167–192.
[50] J., Tits, A ‘theorem of Lie–Kolchin’ for trees. In: Contributions to Algebra: A Collection of Papers Dedicated to Ellis Kolchin, Academic Press, New York, 1977 Google Scholar.
[51] M., Urbański and L. Q., Zamboni, On free actions on Λ-trees, Math. Proc. Camb. Phil. Soc. 113 (1993 Google Scholar), 535–542.
[52] M. J., Wicks, Commutators in free products, J. London Math. Soc. 37 (1962 Google Scholar), 433–444.
[53] M. J., Wicks, A general solution of binary homogeneous equations over free groups, Pacific J. Math. 41 (1972 Google Scholar), 543–561.
[54] D. L., Wilkens, Group actions on trees and length functions, Michigan Math. J. 35 (1988 Google Scholar), 141–150.
[55] A., Zastrow, Construction of an infinitely generated group that is not a free product of surface groups and abelian groups, but which acts freely on an ℝ-tree, Proc. Royal Soc. Edinburgh (A) 128 (1998 Google Scholar), 433–445.

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