Editing Geometric group theory

{{Short description|Area in mathematics devoted to the study of finitely generated groups}}
[[File:F2 Cayley Graph.png|thumb|The [[Cayley graph]] of a [[free group]] with two generators. This is a [[hyperbolic group]] whose [[Gromov boundary]] is a [[Cantor set]]. Hyperbolic groups and their boundaries are important topics in geometric group theory, as are Cayley graphs.]]

'''Geometric group theory''' is an area in [[mathematics]] devoted to the study of [[finitely generated group]]s via exploring the connections between [[algebra]]ic properties of such [[group (mathematics)|groups]] and [[topology|topological]] and [[geometry|geometric]] properties of spaces on which these groups can [[Group action (mathematics)|act]] non-trivially (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 graph]]s of groups, which, in addition to the [[graph (discrete mathematics)|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 [[computational complexity theory|complexity theory]], [[mathematical logic]], the study of [[Lie group]]s 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]] practiced on many occasions, such as teaching mathematics, reciting [[Stéphane Mallarmé|Mallarmé]], or greeting a friend".<ref>P. de la Harpe, [https://books.google.com/books?id=60fTzwfqeQIC&dq=de+la+Harpe,+Topics+in+geometric+group+theory&pg=PP1 ''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}}.</ref>{{rp|3}}

== 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>).

== Modern themes and developments ==
{{Prose|section|date=January 2012}}

Notable themes and developments in geometric group theory in 1990s and 2000s include:

*Gromov's program to study quasi-isometric properties of groups.
:A particularly influential broad theme in the area is [[Mikhail Gromov (mathematician)|Gromov]]'s program<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> of classifying [[Generating set of a group#Finitely generated group|finitely generated groups]] according to their large scale geometry. Formally, this means classifying finitely generated groups with their [[word metric]] up to [[Glossary of Riemannian and metric geometry#Q|quasi-isometry]]. This program involves:
:#The study of properties that are invariant under [[quasi-isometry]]. Examples of such properties of finitely generated groups include: the [[growth rate (group theory)|growth rate]] of a finitely generated group; the [[Dehn function#Isoperimetric function|isoperimetric function]] or [[van Kampen diagram|Dehn function]] of a [[finitely presented group]]; the number of [[End (topology)#Ends of graphs and groups|ends of a group]]; [[hyperbolic group|hyperbolicity of a group]]; the [[homeomorphism]] type of the [[Gromov boundary]] of a hyperbolic group;<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.</ref> [[ultralimit|asymptotic cone]]s of finitely generated groups (see e.g.<ref>{{cite journal |first=Tim R. |last=Riley |title=Higher connectedness of asymptotic cones |journal=Topology |volume=42 |issue=6 |pages=1289–1352 |year=2003 |doi=10.1016/S0040-9383(03)00002-8 |doi-access=free }}</ref><ref>{{cite journal |first1=Linus |last1=Kramer |author2-link=Saharon Shelah |first2=Saharon |last2=Shelah |first3=Katrin |last3=Tent|author3-link= Katrin Tent |first4=Simon |last4=Thomas |title=Asymptotic cones of finitely presented groups |journal=[[Advances in Mathematics]] |volume=193 |issue=1 |pages=142–173 |year=2005 |doi=10.1016/j.aim.2004.04.012 |doi-access=free |arxiv=math/0306420 |s2cid=4769970 }}</ref>); [[Amenable group|amenability]] of a finitely generated group; being virtually [[Abelian group|abelian]] (that is, having an abelian subgroup of finite [[index of a subgroup|index]]); being virtually [[Nilpotent group|nilpotent]]; being virtually [[Free group|free]]; being [[Finitely presented group|finitely presentable]]; being a finitely presentable group with solvable [[Word problem for groups|Word Problem]]; and others.
:#Theorems which use quasi-isometry invariants to prove algebraic results about groups, for example: [[Gromov's theorem on groups of polynomial growth|Gromov's polynomial growth theorem]]; [[Stallings theorem about ends of groups|Stallings' ends theorem]]; [[Mostow rigidity theorem]].
:#Quasi-isometric rigidity theorems, in which one classifies algebraically all groups that are quasi-isometric to some given group or metric space. This direction was initiated by the work of [[Richard Schwartz (mathematician)|Schwartz]] on quasi-isometric rigidity of rank-one lattices<ref>{{cite journal |first=R.E. |last=Schwartz |title=The quasi-isometry classification of rank one lattices |journal=Publications Mathématiques de l'Institut des Hautes Études Scientifiques |volume=82 |issue=1 |pages=133–168 |year=1995 |doi=10.1007/BF02698639 |s2cid=67824718 |url=http://www.numdam.org/item/PMIHES_1995__82__133_0/ }}</ref> and the work of [[Benson Farb]] and Lee Mosher on quasi-isometric rigidity of [[Baumslag–Solitar group]]s.<ref>{{cite journal |first1=Benson |last1=Farb |author1-link=Benson Farb|first2=Lee |last2=Mosher |title=A rigidity theorem for the solvable Baumslag–Solitar groups. With an appendix by Daryl Cooper |journal=[[Inventiones Mathematicae]] |volume=131 |issue=2 |pages=419–451 |year=1998 |doi=10.1007/s002220050210| mr=1608595 |s2cid=121180189 }}</ref>
*The theory of [[hyperbolic group|word-hyperbolic]] and [[Relatively hyperbolic group|relatively hyperbolic]] groups. A particularly important development here is the work of [[Zlil Sela]] in 1990s resulting in the solution of the [[Group isomorphism problem|isomorphism problem]] for word-hyperbolic groups.<ref>{{cite journal |first=Zlil |last=Sela |title=The isomorphism problem for hyperbolic groups. I |journal=[[Annals of Mathematics]] |series=(2) |volume=141  |issue=2 |pages=217–283 |year=1995 |jstor=2118520|mr=1324134|doi=10.2307/2118520}}</ref> The notion of a relatively hyperbolic groups was originally introduced by Gromov in 1987<ref name="M. Gromov, 1987, pp. 75–263"/> and refined by  Farb<ref>{{cite journal |first=Benson |last=Farb |author-link=Benson Farb| title=Relatively hyperbolic groups |journal=[[Geometric and Functional Analysis]] |volume=8 |issue=5 |pages=810–840 |year=1998 |doi=10.1007/s000390050075|mr=1650094 |s2cid=123370926 }}</ref> and [[Brian Bowditch]],<ref>{{cite book |first=Brian H. |last=Bowditch |author-link=Brian Bowditch|title=Treelike Structures Arising from Continua and Convergence Groups |url=https://books.google.com/books?id=95nTCQAAQBAJ |year=1999 |publisher=American Mathematical Society |isbn=978-0-8218-1003-3 |series=Memoirs American Mathematical Society |volume=662}}</ref> in the 1990s. The study of relatively hyperbolic groups gained prominence in the 2000s.
*Interactions with mathematical logic and the study of the first-order theory of free groups. Particularly important progress occurred on the famous [[Free group#Tarski.27s problems|Tarski conjecture]]s, due to the work of Sela<ref>Zlil Sela, ''Diophantine geometry over groups and the elementary theory of free and hyperbolic groups.'' Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), pp. 87–92, Higher Ed. Press, Beijing, 2002.</ref> as well as of [[Olga Kharlampovich]] and Alexei Myasnikov.<ref>{{cite journal |first1=Olga |last1=Kharlampovich |first2=Alexei  |last2=Myasnikov |title=Tarski's problem about the elementary theory of free groups has a positive solution |journal=Electronic Research Announcements of the American Mathematical Society |volume=4 |issue= 14|pages=101–8 |year=1998 |doi=10.1090/S1079-6762-98-00047-X |mr=1662319  |doi-access=free }}</ref> The study of [[limit group]]s and introduction of the language and machinery of [[noncommutative algebraic geometry|non-commutative algebraic geometry]] gained prominence.
*Interactions with computer science, complexity theory and the theory of formal languages. This theme is exemplified by the development of the theory of [[automatic group]]s,<ref>D. B. A. Epstein, J. W. Cannon, D. Holt, S. Levy, M. Paterson, W. Thurston. ''[[Word Processing in Groups]]''. Jones and Bartlett Publishers, Boston, MA, 1992.</ref> a notion that imposes certain geometric and language theoretic conditions on the multiplication operation in a finitely generated group.
*The study of isoperimetric inequalities, Dehn functions and their generalizations for finitely presented group. This includes, in particular, the work of Jean-Camille Birget, Aleksandr Olʹshanskiĭ, [[Eliyahu Rips]] and [[Mark Sapir]]<ref>{{cite journal |first1=Mark |last1=Sapir |author1-link=Mark Sapir|first2=Jean-Camille |last2=Birget |first3=Eliyahu |last3=Rips|author3-link=Eliyahu Rips |title=Isoperimetric and isodiametric functions of groups |journal=[[Annals of Mathematics]] |series= (2) |volume=156 |issue=2 |pages=345–466 |year=2002 |doi=10.2307/3597195 |jstor=3597195|arxiv=math/9811105 |s2cid=119728458 }}</ref><ref>{{cite journal |first1=Jean-Camille |last1=Birget |first2=Aleksandr Yu. |last2= Olʹshanskiĭ |first3=Eliyahu |last3=Rips |author3-link=Eliyahu Rips|first4=Mark |last4=Sapir |author4-link=Mark Sapir| title=Isoperimetric functions of groups and computational complexity of the word problem |journal=[[Annals of Mathematics]] |series= (2) |volume=156 |issue=2 |pages=467–518 |year=2002 |doi=10.2307/3597196 |jstor=3597196 |arxiv=math/9811106 |s2cid=14155715 }}</ref> essentially characterizing the possible Dehn functions of finitely presented groups, as well as results providing explicit constructions of groups with fractional Dehn functions.<ref>{{cite journal |first=M.R. |last=Bridson |title=Fractional isoperimetric inequalities and subgroup distortion |journal=Journal of the American Mathematical Society |volume=12 |issue=4 |pages=1103–18 |year=1999 |doi=10.1090/S0894-0347-99-00308-2  |mr=1678924|s2cid=7981000 |doi-access=free }}</ref>
*The theory of toral or [[JSJ decomposition|JSJ-decompositions]] for [[3-manifold]]s was originally brought into a group theoretic setting by Peter Kropholler.<ref>{{Cite journal|last=Kropholler|first=P. H.|date=1990|title=An Analogue of the Torus Decomposition Theorem for Certain Poincaré Duality Groups|url=https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/plms/s3-60.3.503|journal=Proceedings of the London Mathematical Society|language=en|volume=s3-60|issue=3|pages=503–529|doi=10.1112/plms/s3-60.3.503|issn=1460-244X}}</ref>  This notion has been developed by many authors for both finitely presented and finitely generated groups.<ref>{{cite journal |first1=E. |last1=Rips |first2=Z. |last2=Sela |title=Cyclic splittings of finitely presented groups and the canonical JSJ decomposition |journal=Annals of Mathematics |series=Second Series |volume=146 |issue=1 |pages=53–109 |year=1997 |doi=10.2307/2951832 |jstor=2951832 }}</ref><ref>{{cite journal |first1=M.J. |last1=Dunwoody |first2=M.E. |last2=Sageev |title=JSJ-splittings for finitely presented groups over slender groups |journal=Inventiones Mathematicae |volume=135 |issue=1 |pages=25–44 |year=1999 |doi=10.1007/s002220050278 |bibcode=1999InMat.135...25D |s2cid=16958457 }}</ref><ref>{{cite journal |first1=P. |last1=Scott |first2=G.A. |last2=Swarup |title=Regular neighbourhoods and canonical decompositions for groups |journal=Electronic Research Announcements of the American Mathematical Society |volume=8 |issue= 3|pages=20–28 |year=2002 |doi=10.1090/S1079-6762-02-00102-6  |mr=1928498|doi-access=free }}</ref><ref>{{cite journal |first=B.H. |last=Bowditch |title=Cut points and canonical splittings of hyperbolic groups |journal=Acta Mathematica |volume=180 |issue=2 |pages=145–186 |year=1998 |doi=10.1007/BF02392898 |doi-access=free }}</ref><ref>{{cite journal |first1=K. |last1=Fujiwara |first2=P. |last2=Papasoglu |title=JSJ-decompositions of finitely presented groups and complexes of groups |journal=Geometric and Functional Analysis |volume=16 |issue=1 |pages=70–125 |year=2006 |doi=10.1007/s00039-006-0550-2 |arxiv=math/0507424 |s2cid=10105697 }}</ref>
*Connections with [[geometric analysis]], the study of [[C*-algebras]] associated with discrete groups and of the theory of free probability. This theme is represented, in particular, by considerable progress on the [[Novikov conjecture]] and the [[Baum–Connes conjecture]] and the development and study of related group-theoretic notions such as topological amenability, asymptotic dimension, uniform embeddability into [[Hilbert space]]s, rapid decay property, and so on (see e.g.<ref>{{cite journal |first=G. |last=Yu |title=The Novikov conjecture for groups with finite asymptotic dimension |journal=Annals of Mathematics |series=Second Series |volume=147 |issue=2 |pages=325–355 |year=1998 |doi=10.2307/121011 |jstor=121011 }}</ref><ref>G. Yu. ''The coarse Baum–Connes conjecture for spaces which admit a uniform embedding into Hilbert space.'' Inventiones Mathematicae, vol 139 (2000), no. 1, pp. 201–240.</ref><ref>{{cite journal |first1=I. |last1=Mineyev |first2=G. |last2=Yu |title=The Baum–Connes conjecture for hyperbolic groups |journal=Inventiones Mathematicae |volume=149 |issue=1 |pages=97–122 |year=2002 |doi=10.1007/s002220200214 |arxiv=math/0105086 |bibcode=2002InMat.149...97M |s2cid=7940721 }}</ref>).
*Interactions with the theory of quasiconformal analysis on metric spaces, particularly in relation to [[Cannon's conjecture]] about characterization of hyperbolic groups with [[Gromov boundary]] homeomorphic to the 2-sphere.<ref>{{cite journal |first1=Mario |last1=Bonk |first2=Bruce |last2=Kleiner |title=Conformal dimension and Gromov hyperbolic groups with 2-sphere boundary |journal=[[Geometry & Topology]] |volume=9 |pages=219–246 |year=2005 |arxiv=math/0208135|doi=10.2140/gt.2005.9.219 | doi-access=free |s2cid=786904 }}</ref><ref>Marc  Bourdon and Hervé Pajot. ''Quasi-conformal geometry and hyperbolic geometry.'' Rigidity in dynamics and geometry (Cambridge, 2000), pp. 1–17, Springer, Berlin, 2002.</ref><ref>Mario Bonk, ''Quasiconformal geometry of fractals.'' [[International Congress of Mathematicians]]. Vol. II, pp. 1349–1373, Eur. Math. Soc., Zürich, 2006.</ref>
*[[Finite subdivision rules]], also in relation to [[Cannon's conjecture]].<ref name="finite">{{cite journal |first1=James W. |last1=Cannon |author1-link=James Cannon (mathematician)|first2=William J. |last2=Floyd |author2-link=William Floyd (mathematician)|first3=Walter R. |last3=Parry |title=Finite subdivision rules |journal=Conformal Geometry and Dynamics |volume=5 |issue= 8|pages=153–196 |year=2001 |doi=10.1090/S1088-4173-01-00055-8 |bibcode=2001CGDAM...5..153C |mr=1875951|doi-access=free }}</ref>
*Interactions with [[topological dynamics]] in the contexts of studying actions of discrete groups on various compact spaces and group compactifications, particularly [[convergence group]] methods<ref>P. Tukia. ''Generalizations of Fuchsian and Kleinian groups.'' First European Congress of Mathematics, Vol. II (Paris, 1992), pp. 447–461, Progr. Math., 120, Birkhäuser, Basel, 1994.</ref><ref>{{cite journal |first=Asli |last=Yaman |title=A topological characterisation of relatively hyperbolic groups |journal=[[Crelle's Journal|Journal für die Reine und Angewandte Mathematik]] |volume=566 |pages=41–89 |year=2004 |mr=2039323}}</ref>
*Development of the theory of group actions on [[real tree|<math>\mathbb R</math>-trees]] (particularly the [[Rips machine]]), and its applications.<ref>{{cite journal |author-link=Mladen Bestvina |first1=M. |last1=Bestvina |first2=M. |last2=Feighn |title=Stable actions of groups on real trees |journal=Inventiones Mathematicae |volume=121 |issue=2 |pages=287–321 |year=1995 |doi=10.1007/BF01884300 |bibcode=1995InMat.121..287B |s2cid=122048815 }}</ref>
*The study of group actions on [[CAT(0) space]]s and CAT(0) cubical complexes,<ref name=Bridson99/> motivated by ideas from Alexandrov geometry.
*Interactions with low-dimensional topology and hyperbolic geometry, particularly the study of 3-manifold groups (see, e.g.,<ref>M. Kapovich, ''Hyperbolic manifolds and discrete groups''. Progress in Mathematics, 183. Birkhäuser Boston, Inc., Boston, MA, 2001.</ref>), [[mapping class group]]s of surfaces, [[braid group]]s and [[Kleinian group]]s.
*Introduction of probabilistic methods to study algebraic properties of "random" group theoretic objects (groups, group elements, subgroups, etc.). A particularly important development here is the work of Gromov who used probabilistic methods to prove<ref>M. Gromov. ''Random walk in random groups.'' Geometric and Functional Analysis, vol. 13 (2003), no. 1, pp. 73–146.</ref> the existence of a finitely generated group that is not uniformly embeddable into a Hilbert space. Other notable developments include introduction and study of the notion of [[generic-case complexity]]<ref>{{cite journal |first1=I. |last1=Kapovich |first2=A. |last2=Miasnikov |first3=P. |last3=Schupp |first4=V. |last4=Shpilrain |title=Generic-case complexity, decision problems in group theory, and random walks |journal=Journal of Algebra |volume=264 |issue=2 |pages=665–694 |year=2003 |doi=10.1016/S0021-8693(03)00167-4 |doi-access=free |arxiv=math/0203239 }}</ref> for group-theoretic and other mathematical algorithms and algebraic rigidity results for generic groups.<ref>{{cite journal |first1=I. |last1=Kapovich |first2=P. |last2=Schupp |first3=V. |last3=Shpilrain |title=Generic properties of Whitehead's algorithm and isomorphism rigidity of random one-relator groups |journal=Pacific Journal of Mathematics |volume=223 |issue=1 |pages=113–140 |year=2006 |doi=10.2140/pjm.2006.223.113 |doi-access=free |arxiv=math/0303386 }}</ref>
*The study of [[automata group]]s and [[iterated monodromy group]]s as [[automorphism group|groups of automorphisms]] of infinite rooted trees. In particular, [[Grigorchuk's group]]s of intermediate growth, and their generalizations, appear in this context.<ref>L. Bartholdi, R. I. Grigorchuk and Z. Sunik. ''Branch groups.'' Handbook of algebra, Vol. 3, pp. 989-1112, North-Holland, Amsterdam, 2003.</ref><ref>V. Nekrashevych. ''Self-similar groups.'' Mathematical Surveys and Monographs, 117. American Mathematical Society, Providence, RI, 2005. {{ISBN|0-8218-3831-8}}.</ref>
*The study of measure-theoretic properties of group actions on [[measure space]]s, particularly introduction and development of the notions of [[measure equivalence]] and [[orbit equivalence]], as well as measure-theoretic generalizations of Mostow rigidity.<ref>{{cite journal |first=A. |last=Furman |title=Gromov's measure equivalence and rigidity of higher rank lattices |journal=Annals of Mathematics |series=Second Series |volume=150 |issue=3 |pages=1059–81 |year=1999 |doi=10.2307/121062 |jstor=121062|arxiv=math/9911262 |bibcode=1999math.....11262F |s2cid=15408706 }}</ref><ref>{{cite journal |first1=N. |last1=Monod |first2=Y. |last2=Shalom |title=Orbit equivalence rigidity and bounded cohomology |journal=Annals of Mathematics |series=Second Series |volume=164 |issue=3 |pages=825–878 |year=2006 |jstor=20160009 |doi=10.4007/annals.2006.164.825|doi-access=free }}</ref>
*The study of unitary representations of discrete groups and [[Kazhdan's property (T)]]<ref>Y. Shalom. ''The algebraization of Kazhdan's property (T).'' International Congress of Mathematicians. Vol. II, pp. 1283–1310, Eur. Math. Soc., Zürich, 2006.</ref>
*The study of ''Out''(''F''<sub>''n''</sub>) (the [[outer automorphism group]] of a [[free group]] of rank ''n'') and of individual automorphisms of free groups. Introduction and the study of Culler-Vogtmann's [[outer space (group theory)|outer space]]<ref>{{cite journal |first1=M. |last1=Culler |author2-link=Karen Vogtmann |first2=K. |last2=Vogtmann |title=Moduli of graphs and automorphisms of free groups |journal=[[Inventiones Mathematicae]] |volume=84 |issue=1 |pages=91–119 |year=1986 |doi=10.1007/BF01388734 |bibcode=1986InMat..84...91C |s2cid=122869546 }}</ref> and of the theory of [[train track (mathematics)|train tracks]]<ref>{{cite journal |first1=Mladen |last1=Bestvina |first2=Michael |last2=Handel |title=Train tracks and automorphisms of free groups |journal=[[Annals of Mathematics]]|series= 2 |volume=135 |issue=1 |pages=1–51 |year=1992 |doi=10.2307/2946562 |jstor=2946562|mr=1147956 }}</ref> for free group automorphisms played a particularly prominent role here.
*Development of [[Bass–Serre theory]], particularly various accessibility results<ref>{{cite journal |first=M.J. |last=Dunwoody |title=The accessibility of finitely presented groups |journal=[[Inventiones Mathematicae]] |volume=81 |issue=3 |pages=449–457 |year=1985 |doi=10.1007/BF01388581 |bibcode=1985InMat..81..449D |s2cid=120065939 }}</ref><ref>{{cite journal |first1=M. |last1=Bestvina |first2=M. |last2=Feighn |title=Bounding the complexity of simplicial group actions on trees |journal=[[Inventiones Mathematicae]] |volume=103 |issue=3 |pages=449–469 |year=1991 |doi=10.1007/BF01239522 |bibcode=1991InMat.103..449B |s2cid=121136037 }}</ref><ref>{{cite journal |first=Zlil  |last=Sela |title=Acylindrical accessibility for groups |journal=[[Inventiones Mathematicae]] |volume=129 |issue=3 |pages=527–565 |year=1997 |doi=10.1007/s002220050172 |bibcode=1997InMat.129..527S |s2cid=122548154 }}</ref> and the theory of tree lattices.<ref>[[Hyman Bass]] and [[Alexander Lubotzky]]. ''Tree lattices. With appendices by Hyman Bass, Lisa Carbone, Alexander Lubotzky, G. Rosenberg and [[Jacques Tits]].'' Progress in Mathematics, 176. Birkhäuser Boston, Inc., Boston, MA, 2001. {{ISBN|0-8176-4120-3}}.</ref> Generalizations of Bass–Serre theory such as the theory of complexes of groups.<ref name=Bridson99>{{harvnb|Bridson|Haefliger|1999}}</ref>
*The study of [[random walk]]s on groups and related boundary theory, particularly the notion of [[Poisson boundary]] (see e.g.<ref>{{cite journal |first=V.A. |last=Kaimanovich |title=The Poisson formula for groups with hyperbolic properties |journal=[[Annals of Mathematics]] |series=2 |volume=152 |issue=3 |pages=659–692 |year=2000 |doi=10.2307/2661351 |jstor=2661351 |arxiv=math/9802132 |s2cid=14774503 }}</ref>). The study of [[Amenable group|amenability]] and of groups whose amenability status is still unknown.
*Interactions with finite group theory, particularly progress in the study of [[subgroup growth]].<ref>[[Alexander Lubotzky]] and Dan Segal. ''Subgroup growth.'' Progress in Mathematics, 212. [[Birkhäuser|Birkhäuser Verlag]], Basel, 2003. {{ISBN|3-7643-6989-2}}. {{MR|1978431}}</ref>
*Studying subgroups and lattices in [[linear group]]s, such as <math>SL(n, \mathbb R)</math>, and of other Lie groups, via geometric methods (e.g. [[Building (mathematics)|buildings]]), [[Algebraic geometry|algebro-geometric]] tools (e.g. [[algebraic group]]s and representation varieties), analytic methods (e.g. unitary representations on Hilbert spaces) and arithmetic methods.
*[[Group cohomology]], using algebraic and topological methods, particularly involving interaction with [[algebraic topology]] and the use of [[Morse theory|morse-theoretic]] ideas in the combinatorial context; large-scale, or coarse (see e.g.<ref>{{cite journal |first1=Mladen |last1=Bestvina|author1-link=Mladen Bestvina |first2=Michael |last2=Kapovich |first3=Bruce |last3=Kleiner |title=Van Kampen's embedding obstruction for discrete groups |journal=[[Inventiones Mathematicae]] |volume=150 |issue=2 |pages=219–235 |year=2002 |doi=10.1007/s00222-002-0246-7 |arxiv=math/0010141|bibcode=2002InMat.150..219B|s2cid=7153145|mr=1933584}}</ref>) homological and cohomological methods.
*Progress on traditional combinatorial group theory topics, such as the [[Burnside problem]],<ref>{{cite journal |first=S.V. |last=Ivanov |title=The free Burnside groups of sufficiently large exponents |journal=[[International Journal of Algebra and Computation]] |volume=4 |issue=1n2 |pages=1–309 |year=1994 |doi=10.1142/S0218196794000026 }}</ref><ref>{{cite journal |first=I.G. |last=Lysënok |title=Infinite Burnside groups of even exponent |journal=[[Izvestiya: Mathematics]] |volume=60 |issue=3 |pages=453–654 |year=1996 |doi=10.1070/im1996v060n03abeh000077 |bibcode=1996IzMat..60..453L |s2cid=250838960 }}</ref> the study of [[Coxeter group]]s and [[Artin group]]s, and so on (the methods used to study these questions currently are often geometric and topological).

== Examples ==

The following examples are often studied in geometric group theory:
{{div col|colwidth=25em}}
* [[Amenable group]]s
* [[Burnside group|Free Burnside groups]]
* The infinite [[cyclic group]] '''[[integer|Z]]'''
* [[Free group]]s
* [[Free product]]s
* [[Outer automorphism group]]s [[Out(Fn)|Out(F<sub>''n''</sub>)]] (via [[Outer space (group theory)|outer space]])
* [[Hyperbolic group]]s
* [[Mapping class group]]s (automorphisms of surfaces)
* [[Symmetric group]]s
* [[Braid group]]s
* [[Coxeter group]]s
* General [[Artin group]]s
* [[Thompson groups|Thompson's group]] ''F''
* [[CAT(0) group]]s
* [[Arithmetic group]]s
* [[Automatic group]]s 
* [[Fuchsian group]]s, [[Kleinian group]]s, and other groups acting properly discontinuously on symmetric spaces, in particular [[Lattice (discrete subgroup)|lattices]] in semisimple Lie groups.
* [[Wallpaper group]]s
* [[Baumslag–Solitar group]]s
* [[Graph of groups|Fundamental groups of graphs of groups]]
* [[Grigorchuk group]]
{{div col end}}

== See also ==
* The [[ping-pong lemma]], a useful way to exhibit a group as a free product
* [[Amenable group]]
* [[Nielsen transformation]]
* [[Tietze transformation]]

== References ==
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{{Reflist|2}}

=== Books and monographs ===

These texts cover geometric group theory and related topics.

*{{cite book|first=Brian H. | last=Bowditch| author-link= Brian Bowditch |title=A course on geometric group theory|series= MSJ Memoirs|volume= 16| publisher=[[Mathematical Society of Japan]] |location= Tokyo|year= 2006|isbn=4-931469-35-3}}
*{{cite book|first1=Martin R.|last1= Bridson|author-link1=Martin Bridson|first2= André |last2= Haefliger |author-link2=André Haefliger| title=Metric spaces of non-positive curvature|series= Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]|volume = 319 |publisher= Springer-Verlag|location= Berlin|year= 1999|isbn=3-540-64324-9 }}
*{{cite book |first1=Michel |last1=Coornaert |first2=Thomas |last2=Delzant |first3=Athanase |last3=Papadopoulos |title=Géométrie et théorie des groupes : les groupes hyperboliques de Gromov |publisher=Springer-Verlag |series=Lecture Notes in Mathematics |volume=1441 |year=1990 |isbn=3-540-52977-2 |mr=1075994}}

*{{cite book | last1=Clay| first1=Matt|last2=Margalit|first2=Dan|year=2017|title=Office Hours with a Geometric Group Theorist |publisher=Princeton University Press|isbn=978-0-691-15866-2}}

*{{cite book |first1=Michel |last1=Coornaert |first2=Athanase |last2=Papadopoulos |title=Symbolic dynamics and hyperbolic groups |publisher=Springer-Verlag |series=Lecture Notes in Mathematics |volume=1539 |year=1993 |isbn=3-540-56499-3 }}
*{{cite book |first=P. |last=de la Harpe |title=Topics in geometric group theory |series=Chicago Lectures in Mathematics |publisher=University of Chicago Press |year=2000 |isbn=0-226-31719-6 }}
*{{cite book|first1= Cornelia|last1=Druţu|author1-link= Cornelia Druțu |first2= Michael|last2=Kapovich |author-link2=Michael Kapovich|title=Geometric Group Theory|series=American Mathematical Society Colloquium Publications|volume= 63| publisher=[[American Mathematical Society]] |year=2018|url=https://www.math.ucdavis.edu/~kapovich/EPR/ggt.pdf|isbn= 978-1-4704-1104-6|mr=3753580}}
*{{cite book |first1=D.B.A. |last1=Epstein |first2=J.W. |last2=Cannon |first3=D. |last3=Holt |first4=S. |last4=Levy |first5=M. |last5=Paterson |first6=W. |last6=Thurston |title= Word Processing in Groups|title-link= Word Processing in Groups |publisher=Jones and Bartlett |year=1992 |isbn=0-86720-244-0 }}
*{{cite book |first=M. |last=Gromov |chapter=Hyperbolic Groups |editor-first=G.M. |editor-last=Gersten |title=Essays in Group Theory |publisher=MSRI |volume=8 |year=1987 |isbn=0-387-96618-8 |pages=75–263 }}
*{{cite conference |first=Mikhael |last=Gromov |title=Asymptotic invariants of infinite groups |editor-first=G.A. |editor-last=Niblo |editor2-first=M.A. |editor2-last=Roller |book-title=Geometric Group Theory: Proceedings of the Symposium held in Sussex 1991 |volume=2 |url=https://books.google.com/books?id=dH02YAfVqkYC&pg=PP1 |date=1993 |publisher=Cambridge University Press |isbn=978-0-521-44680-8 |pages=1–295 | series=London Mathematical Society Lecture Note Series}}
*{{cite book |first=M. |last=Kapovich |title=Hyperbolic Manifolds and Discrete Groups |series=Progress in Mathematics |volume=183 |url=https://books.google.com/books?id=YmphheDo18kC |year=2001 |publisher=Birkhäuser |isbn=978-0-8176-3904-4}}
*{{cite book |author-link=Roger Lyndon |first1=Roger C. |last1=Lyndon |first2=Paul E. |last2=Schupp |title=Combinatorial Group Theory |url=https://books.google.com/books?id=cOLrCAAAQBAJ |orig-year=1977 |date=2015 |publisher=Springer |isbn=978-3-642-61896-3 |series=Classics in mathematics}}
*{{cite book |first=A.Yu. |last=Ol'shanskii |title=Geometry of Defining Relations in Groups |url=https://books.google.com/books?id=uS_pCAAAQBAJ&pg=PP1 |date=2012 |publisher=Springer |orig-year=1991 |isbn=978-94-011-3618-1 }}
*{{cite book |first=John |last=Roe |title=Lectures on Coarse Geometry |url=https://books.google.com/books?id=jbsFCAAAQBAJ |year=2003 |series=University Lecture Series |volume=31 |publisher=American Mathematical Society |isbn=978-0-8218-3332-2}}

== External links ==
*[http://www.math.ucsb.edu/~mccammon/geogrouptheory/ Jon McCammond's Geometric Group Theory Page]
*[http://www.math.mcgill.ca/wise/ggt/cayley.html ''What is Geometric Group Theory?'' By Daniel Wise]
*[https://web.archive.org/web/20040830075241/http://zebra.sci.ccny.cuny.edu/web/nygtc/problems/ Open Problems in combinatorial and geometric group theory]
*[http://xstructure.inr.ac.ru/x-bin/theme3.py?level=1&index1=-98867 Geometric group theory Theme on arxiv.org]

[[Category:Geometric group theory| ]]
[[Category:Group theory]]