Editing Geometric group theory (section)

== History ==

Geometric group theory grew out of [[combinatorial group theory]] that largely studied properties of [[discrete group]]s via analyzing [[Presentation of a group|group presentations]], which describe groups as [[quotient group|quotients]] of [[free group]]s; this field was first systematically studied by [[Walther von Dyck]], student of [[Felix Klein]], in the early 1880s,<ref name="stillwell374">{{Citation
| publisher = Springer
| isbn = 978-0-387-95336-6
| last = Stillwell
| first = John
| title = Mathematics and its history
| year = 2002
| page = [https://books.google.com/books?id=WNjRrqTm62QC&pg=PA374 374]
}}</ref> 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 theory|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 [[Max Dehn]], [[Jakob Nielsen (mathematician)|Jakob Nielsen]], [[Kurt Reidemeister]] and [[Otto Schreier]], [[J. H. C. Whitehead]], [[Egbert van Kampen]], amongst others, introduced some topological and geometric ideas into the study of discrete groups.<ref>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.</ref> 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<ref>{{cite journal |first=Martin |last=Greendlinger |title=Dehn's algorithm for the word problem |journal=Communications on Pure and Applied Mathematics |volume=13 |issue=1 |pages=67–83 |year=1960 |doi=10.1002/cpa.3160130108 }}</ref><ref>{{cite journal |first=Martin |last=Greendlinger |title=An analogue of a theorem of Magnus |journal=Archiv der Mathematik |volume=12 |issue=1 |pages=94–96 |year=1961 |doi=10.1007/BF01650530 |s2cid=120083990 }}</ref> and further developed by [[Roger Lyndon]] and [[Paul Schupp]].<ref>[[Roger Lyndon]] and [[Paul Schupp]], [https://books.google.com/books?id=aiPVBygHi_oC&q=lyndon+and+schupp ''Combinatorial Group Theory''], Springer-Verlag, Berlin, 1977. Reprinted in the "Classics in mathematics" series, 2000.</ref> It studies [[van Kampen diagram]]s, 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,<ref>J.-P. Serre, ''Trees''. Translated from the 1977 French original by [[John Stillwell]]. Springer-Verlag, Berlin-New York, 1980. {{ISBN|3-540-10103-9}}.</ref> derives structural algebraic information about groups by studying group actions on [[Tree (graph theory)|simplicial trees]].
External precursors of geometric group theory include the study of lattices in Lie groups, especially [[Mostow's rigidity theorem]], the study of [[Kleinian group]]s, and the progress achieved in [[low-dimensional topology]] and hyperbolic geometry in the 1970s and early 1980s, spurred, in particular, by [[William Thurston]]'s [[Geometrization conjecture|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 [[Mikhail Gromov (mathematician)|Mikhail Gromov]] ''"Hyperbolic groups"''<ref name="M. Gromov, 1987, pp. 75–263">Mikhail Gromov, ''Hyperbolic Groups'', in "Essays in Group Theory" (Steve M. Gersten, ed.), MSRI Publ. 8, 1987, pp. 75–263.</ref> 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'',<ref>Mikhail 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.</ref> that outlined Gromov's program of understanding discrete groups up to [[Glossary of Riemannian and metric geometry#Q|quasi-isometry]]. The work of Gromov had a transformative effect on the study of discrete groups<ref>Iliya Kapovich and Nadia 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 [in 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)."</ref><ref>[[Brian 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 [Gr]."</ref><ref>{{cite journal |first=Gabor |last=Elek |title=The mathematics of Misha Gromov |journal=[[Acta Mathematica Hungarica]] |volume=113 |issue=3 |pages=171–185 |year=2006 |doi=10.1007/s10474-006-0098-5 |doi-access=free |s2cid=120667382 |quote=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."}}</ref> and the phrase "geometric group theory" started appearing soon afterwards. (see e.g.<ref>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}}.</ref>).