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Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act (that is, when the groups in question are realized as geometric symmetries or continuous transformations of some spaces). Another important idea in geometric group theory is to consider finitely generated groups themselves as geometric objects. This is usually done by studying the Cayley graphs of groups, which, in addition to the graph structure, are endowed with the structure of a metric space, given by the so-called word metric. Geometric group theory, as a distinct area, is relatively new, and became a clearly identifiable branch of mathematics in the late 1980s and early 1990s. Geometric group theory closely interacts with low-dimensional topology, hyperbolic geometry, algebraic topology, computational group theory and differential geometry. There are also substantial connections with complexity theory, mathematical logic, the study of Lie Groups and their discrete subgroups, dynamical systems, probability theory, K-theory, and other areas of mathematics. In the introduction to his book ''Topics in Geometric Group Theory'', Pierre de la Harpe wrote: "One of my personal beliefs is that fascination with symmetries and groups is one way of coping with frustrations of life's limitations: we like to recognize symmetries which allow us to recognize more than what we can see. In this sense the study of geometric group theory is a part of culture, and reminds me of several things that Georges de Rham practices on many occasions, such as teaching mathematics, reciting Mallarmé, or greeting a friend" (page 3 in 〔P. de la Harpe, (''Topics in geometric group theory''. ) Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 2000. ISBN 0-226-31719-6, ISBN 0-226-31721-8.〕). == History == Geometric group theory grew out of combinatorial group theory that largely studied properties of discrete groups via analyzing group presentations, that describe groups as quotients of free groups; this field was first systematically studied by Walther von Dyck, student of Felix Klein, in the early 1880s, while an early form is found in the 1856 icosian calculus of William Rowan Hamilton, where he studied the icosahedral symmetry group via the edge graph of the dodecahedron. Currently combinatorial group theory as an area is largely subsumed by geometric group theory. Moreover, the term "geometric group theory" came to often include studying discrete groups using probabilistic, measure-theoretic, arithmetic, analytic and other approaches that lie outside of the traditional combinatorial group theory arsenal. In the first half of the 20th century, pioneering work of Dehn, Nielsen, Reidemeister and Schreier, Whitehead, van Kampen, amongst others, introduced some topological and geometric ideas into the study of discrete groups.〔Bruce Chandler and Wilhelm Magnus. ''The history of combinatorial group theory. A case study in the history of ideas.'' Studies in the History of Mathematics and Physical Sciences, vo. 9. Springer-Verlag, New York, 1982.〕 Other precursors of geometric group theory include small cancellation theory and Bass–Serre theory. Small cancellation theory was introduced by Martin Grindlinger in the 1960s〔M. Greendlinger, (''Dehn's algorithm for the word problem.'' ) Communications in Pure and Applied Mathematics, vol. 13 (1960), pp. 67–83.〕〔M. Greendlinger, ''An analogue of a theorem of Magnus''. Archiv der Mathematik, vol. 12 (1961), pp. 94–96.〕 and further developed by Roger Lyndon and Paul Schupp.〔R. Lyndon and P. Schupp, (''Combinatorial Group Theory'' ), Springer-Verlag, Berlin, 1977. Reprinted in the "Classics in mathematics" series, 2000.〕 It studies van Kampen diagrams, corresponding to finite group presentations, via combinatorial curvature conditions and derives algebraic and algorithmic properties of groups from such analysis. Bass–Serre theory, introduced in the 1977 book of Serre,〔J.-P. Serre, ''Trees''. Translated from the 1977 French original by John Stillwell. Springer-Verlag, Berlin-New York, 1980. ISBN 3-540-10103-9.〕 derives structural algebraic information about groups by studying group actions on simplicial trees. External precursors of geometric group theory include the study of lattices in Lie Groups, especially Mostow rigidity theorem, the study of Kleinian groups, and the progress achieved in low-dimensional topology and hyperbolic geometry in the 1970s and early 1980s, spurred, in particular, by Thurston's Geometrization program. The emergence of geometric group theory as a distinct area of mathematics is usually traced to the late 1980s and early 1990s. It was spurred by the 1987 monograph of Gromov ''"Hyperbolic groups"''〔M. Gromov, ''Hyperbolic Groups'', in "Essays in Group Theory" (G. M. Gersten, ed.), MSRI Publ. 8, 1987, pp. 75–263.〕 that introduced the notion of a hyperbolic group (also known as ''word-hyperbolic'' or ''Gromov-hyperbolic'' or ''negatively curved'' group), which captures the idea of a finitely generated group having large-scale negative curvature, and by his subsequent monograph ''Asymptotic Invariants of Infinite Groups'',〔M. Gromov, ''"Asymptotic invariants of infinite groups"'', in "Geometric Group Theory", Vol. 2 (Sussex, 1991), London Mathematical Society Lecture Note Series, 182, Cambridge University Press, Cambridge, 1993, pp. 1–295.〕 that outlined Gromov's program of understanding discrete groups up to quasi-isometry. The work of Gromov had a transformative effect on the study of discrete groups〔I. Kapovich and N. Benakli. ''Boundaries of hyperbolic groups.'' Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001), pp. 39–93, Contemp. Math., 296, Amer. Math. Soc., Providence, RI, 2002. From the Introduction:" In the last fifteen years geometric group theory has enjoyed fast growth and rapidly increasing influence. Much of this progress has been spurred by remarkable work of M. L. Gromov (Essays in group theory, 75–263, Springer, New York, 1987; in Geometric group theory, Vol. 2 (Sussex, 1991), 1–295, Cambridge Univ. Press, Cambridge, 1993 ), who has advanced the theory of word-hyperbolic groups (also referred to as Gromov-hyperbolic or negatively curved groups)."〕〔B. H. Bowditch, ''Hyperbolic 3-manifolds and the geometry of the curve complex.'' European Congress of Mathematics, pp. 103–115, Eur. Math. Soc., Zürich, 2005. From the Introduction:" Much of this can be viewed in the context of geometric group theory. This subject has seen very rapid growth over the last twenty years or so, though of course, its antecedents can be traced back much earlier. () The work of Gromov has been a major driving force in this. Particularly relevant here is his seminal paper on hyperbolic groups ()."〕〔G. Elek. ''The mathematics of Misha Gromov.'' Acta Mathematica Hungarica, vol. 113 (2006), no. 3, pp. 171–185. From p. 181: "Gromov's pioneering work on the geometry of discrete metric spaces and his quasi-isometry program became the locomotive of geometric group theory from the early eighties."〕 and the phrase "geometric group theory" started appearing soon afterwards. (see, e.g.,〔Geometric group theory. Vol. 1. Proceedings of the symposium held at Sussex University, Sussex, July 1991. Edited by Graham A. Niblo and Martin A. Roller. London Mathematical Society Lecture Note Series, 181. Cambridge University Press, Cambridge, 1993. ISBN 0-521-43529-3.〕). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Geometric group theory」の詳細全文を読む スポンサード リンク
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