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[[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.]]
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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 [[topology|topological]] and [[geometry|geometric]] properties of spaces on which these groups [[Group action|act]] (that is, when the groups in question are realized as geometric symmetries or continuous transformations of some spaces).
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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 structure, are endowed with the structure of a [[metric space]], given by the so-called [[word metric]].
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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.
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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 [[Stéphane Mallarmé|Mallarmé]], or greeting a friend" (page 3 in <ref>P. de la Harpe, [http://books.google.com/books?id=60fTzwfqeQIC&pg=PP1&dq=de+la+Harpe,+Topics+in+geometric+group+theory ''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>).
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== History ==
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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]], that 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
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}}</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-theoretic, arithmetic, analytic and other approaches that lie outside of the traditional combinatorial group theory arsenal.
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In the first half of the 20th century, pioneering work of [[Max Dehn|Dehn]], [[Jakob Nielsen (mathematician)|Nielsen]], [[Kurt Reidemeister|Reidemeister]] and [[Otto Schreier|Schreier]], [[J. H. C. Whitehead|Whitehead]], [[Egbert van Kampen|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&ndash;Serre theory]].
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Small cancellation theory was introduced by [[Martin Grindlinger]] in 1960s<ref>M. Greendlinger, [http://www3.interscience.wiley.com/journal/113397463/abstract?CRETRY=1&SRETRY=0 ''Dehn's algorithm for the word problem.''] Communications in Pure and Applied Mathematics, vol. 13 (1960), pp. 67–83.</ref><ref>M. Greendlinger, ''An analogue of a theorem of Magnus''. Archiv der Mathematik, vol. 12 (1961), pp. 94–96.</ref> and further developed by [[Roger Lyndon]] and [[Paul Schupp]].<ref>[[Roger Lyndon|R. Lyndon]] and P. Schupp, [http://books.google.com/books?id=aiPVBygHi_oC&printsec=frontcover&dq=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&ndash;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 rigidity theorem]], the study of [[Kleinian group]]s, and the progress achieved in [[low-dimensional topology]] and hyperbolic geometry in 1970s and early 1980s, spurred, in particular, by [[William Thurston|Thurston's]] [[Geometrization conjecture|Geometrization program]].


The emergence of geometric group theory as a distinct area of mathematics is usually traced to late 1980s and early 1990s. It was spurred by the 1987 monograph of [[Mikhail Gromov (mathematician)|Gromov]] ''"Hyperbolic groups"''<ref name="M. Gromov, 1987, pp. 75–263">M. Gromov, ''Hyperbolic Groups'', in "Essays in Group Theory" (G. 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 Inifinite Groups'',<ref>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.</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>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 [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>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 [Gr]."</ref><ref>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."</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>).
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== Modern themes and developments ==
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*Gromov's program to study quasi-isometric properties of groups.
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:A particularly influential broad theme in the area is [[Mikhail Gromov (mathematician)|Gromov]]'s program<ref>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.</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 ends of a group; [[hyperbolic group|hyperbolicity of a group]]; the [[homeomorphism]] type of the [[Gromov boundary]] of a hyperbolic group;<ref>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.</ref> [[ultralimit|asymptotic cone]]s of finitely generated groups (see, e.g.,<ref>T. R. Riley, [http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V1J-48173YV-2&_user=10&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=836106f8cf958990dfd27ab111c1286a ''Higher connectedness of asymptotic cones.''] [[Topology (journal)|Topology]], vol. 42 (2003), no. 6, pp. 1289–1352.</ref><ref>L. Kramer, S. Shelah, K. Tent and S. Thomas. [http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6W9F-4CSG3HS-1&_user=10&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=6ba86760e3a9331e0b330a291a0cf444 ''Asymptotic cones of finitely presented groups.''] Advances in Mathematics, vol. 193 (2005), no. 1, pp. 142–173.</ref>); [[amenability]] of a finitely generated group; being virtually [[Abelian group|abelian]] (that is, having an abelian subgroup of finite 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' 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|Schwartz]] on quasi-isometric rigidity of rank-one lattices<ref>R. E. Richard. ''The quasi-isometry classification of rank one lattices.'' Institut des Hautes Études Scientifiques. Publications Mathématiques. No. 82 (1995), pp. 133-168.</ref> and the work of Farb and Mosher on quasi-isometric rigidity of [[Baumslag-Solitar group]]s.<ref>B. Farb and L. Mosher. ''A rigidity theorem for the solvable Baumslag-Solitar groups. With an appendix by Daryl Cooper.'' [[Inventiones Mathematicae]], vol. 131 (1998), no. 2, pp. 419–451.</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|Sela]] in 1990s resulting in the solution of the [[isomorphism problem (groups)|isomorphism problem]] for word-hyperbolic groups.<ref>Z. Sela, [http://www.jstor.org/pss/2118520 ''The isomorphism problem for hyperbolic groups. I.''] [[Annals of Mathematics]] (2), vol. 141 (1995), no. 2, pp. 217–283.</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>B. Farb. ''Relatively hyperbolic groups.'' Geometric and Functional Analysis, vol. 8 (1998), no. 5, pp. 810–840.</ref> and [[Brian Bowditch|Bowditch]],<ref>B. H. Bowditch. ''Treelike structures arising from continua and convergence groups.'' Memoirs American Mathematical Society vol. 139 (1999), no. 662.</ref> in the 1990s. The study of relatively hyperbolic groups gained prominence in 2000s.
*Interactions with mathematical logic and the study of 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>Z.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 Kharlampovich and Myasnikov.<ref>O. Kharlampovich and A. Myasnikov, Tarski's problem about the elementary theory of free groups has a positive solution. Electronic Research Announcements of the American Mathematical Society, vol. 4 (1998), pp. 101–108.</ref> The study of [[limit group]]s and introduction of the language and machinery of 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 generate group.
*The study of isoperimetric inequalities, Dehn functions and their generalizations for finitely presented group. This includes, in particular, the work of Birget, Ol'shanskii, [[Eliyahu Rips|Rips]] and Sapir<ref>M. Sapir, J.-C. Birget, E. Rips, ''Isoperimetric and isodiametric functions of groups.''
[[Annals of Mathematics]] (2), vol 156 (2002), no. 2, pp. 345–466.</ref><ref>J.-C. Birget, A. Yu. Ol'shanskii, E. Rips, M. Sapir, ''Isoperimetric functions of groups and computational complexity of the word problem.''
[[Annals of Mathematics]] (2), vol 156 (2002), no. 2, pp. 467-518.</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>M. R. Bridson, ''Fractional isoperimetric inequalities and subgroup distortion.'' [[Journal of the American Mathematical Society]], vol. 12 (1999), no. 4, pp. 1103–1118.</ref>
*Development of the theory of JSJ-decompositions for finitely generated and finitely presented groups.<ref>E. Rips and Z. Sela, ''Cyclic splittings of finitely presented groups and the canonical JSJ decomposition.'' Annals of Mathematics (2), vol. 146 (1997), no. 1, pp. 53–109.</ref><ref>M. J. Dunwoody and M. E. Sageev. ''JSJ-splittings for finitely presented groups over slender groups.'' [[Inventiones Mathematicae]], vol. 135 (1999), no. 1, pp. 25–44.</ref><ref>P. Scott and G. A. Swarup. ''Regular neighbourhoods and canonical decompositions for groups.'' Electronic Research Announcements of the American Mathematical Society, vol. 8 (2002), pp. 20–28.</ref><ref>B. H. Bowditch. ''Cut points and canonical splittings of hyperbolic groups.'' [[Acta Mathematica]], vol. 180 (1998), no. 2, pp. 145–186.</ref><ref>K. Fujiwara and P. Papasoglu, ''JSJ-decompositions of finitely presented groups and complexes of groups.'' Geometric and Functional Analysis, vol. 16 (2006), no. 1, pp. 70-125.</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 spaces, rapid decay property, and so on (see, for example,<ref>G. Yu. ''The Novikov conjecture for groups with finite asymptotic dimension.'' Annals of Mathematics (2), vol. 147 (1998), no. 2, pp. 325–355.</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>I. Mineyev and G. Yu. ''The Baum–Connes conjecture for hyperbolic groups.'' [[Inventiones Mathematicae]], vol. 149 (2002), no. 1, pp. 97–122.</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>M. Bonk and B. Kleiner. ''Conformal dimension and Gromov hyperbolic groups with 2-sphere boundary.'' [[Geometry and Topology]], vol. 9 (2005), pp. 219–246.</ref><ref>M. Bourdon and H. Pajot. ''Quasi-conformal geometry and hyperbolic geometry.'' Rigidity in dynamics and geometry (Cambridge, 2000), pp. 1–17, Springer, Berlin, 2002.</ref><ref>M. 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">J. W. Cannon, W. J. Floyd, W. R. Parry. ''Finite subdivision rules''. Conformal Geometry and Dynamics, vol. 5 (2001), pp. 153&ndash;196.</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>A. Yaman. ''A topological charactesization of relatively hyperbolic groups.'' Journal für die Reine und Angewandte Mathematik, vol. 566 (2004), pp. 41–89.</ref>
The IndependentIn the last 2 hours Obama's day: Raising money, talking immigration in SF  USA TODAY google US (Today) San Francisco ChronicleObama's day: Raising money, talking immigration in SFUSA TODAYPresident Obama continues his West Coast trip Monday, traveling to San Francisco [http://www.romavilla.com/custom/ordersummary.asp Mbt Shoes] to raise campaign money and talk about immigration. Obama challenges immigration heckler  Politico Obama heckled at immigration speech on ending deportations  BBC News Obama Immigration Speech Gets Heated Input  Wall Street Journal CPS: 'Insufficient evidence to show officer at gate lied' ITV News google UK (Today) Leamington ObserverCPS: 'Insufficient evidence to show officer at gate lied'ITV NewsWe have considered all of the evidence in this case, including previously unseen, [http://www.romagnolabeefgenetics.com.au/common/standard.asp Nike Air Max] unedited CCTV footage from Downing Street, not known by the media. <br><br>Ten months later the pharmaceutical industry announced it's updating its voluntary standards for directtoconsumer advertising. It pledged to stop using actors to play doctors, and to make sure endorsers who said they had used a particular drug had actually done so.. <br><br>I can review your current insurance coverages to make sure they meet your needs. The checkup includes a test of all your State Farm coverages and it's FREE. Celebrity gossip can boost selfesteem. Individuals with low selfesteem can get a boost by identifying with and aspiring to some celebrity ideal. <br><br>Mr. Blake's photographs are excellent. He joined the Air Force when WWII started and was posted overseas to York, England where he met the romance of his life, Connie. Following the war they moved to Edmonton where Jack worked like a sheet metal aircraft technician for Northwest Industries. <br><br>Can be reached by car in about an hour. Call Robin Schoonmaker for more details on [http://www.gregoryburgessarchitects.com.au/Mobile/MasterPages/header.asp Air Max 2014] living in South Jersey!. Educate your reader. Don't assume people know. It all worked out perfectly. God's timing was better than anything I could have forced. Efficiencies gained throughtechnology applications means CAPC to add capacity and to monitorcompliance with quality metrics. Third party payers (Medicaid and private insurance payers) are increasing accountability to providers to produce high quality costeffective medical care. <br><br>(CANOGA PARK, CA) free aunt nephew incest pics  There is really a government allegation that a performer referred to as Seduction, whose real name may be Mieke Jackson, was underage while performing. Improved linkages Loose Aunt Nephew Incest Pics between research, policy, and practice; Chronic Myelogenous Leukemia (CML) free aunt nephew incest pics beyond the Free Aunt Nephew Incest Pics scenes we all know to connect with people, places, ideas, and 2005 Adult Day Services Week: Services, Integration Complimentary Gay Mpeg and CommunityThe naked rambler is free of charge aunt nephew incest pics jailed for 2 weeks was expanded to 13 sites in the Hyderabad /RangaReddy Detached Aunt Nephew Incest Pics region of Andhra Site free aunt nephew incest pics last revised: September 2005 more from our Patron, the [http://www.nadya.com.au/flux/core/styles/library.asp Ray Ban Sunglasses] President of Ireland Relieve Aunt Nephew Incest Pics An interesting new concept is working its way "free aunt nephew incest pics" with the adult indust.<ul>
*Development of the theory of group actions on [[real tree|<math>\mathbb R</math>-trees]] (particularly the Rips machine), and its applications.<ref>[[Mladen Bestvina|M. Bestvina]] and M. Feighn. ''Stable actions of groups on real trees.'' [[Inventiones Mathematicae]], vol. 121 (1995), no. 2, pp. 287–321.</ref>
 
*The study of group actions on [[CAT(0) space]]s and CAT(0) cubical complexes,<ref>M. R. Bridson and [[A. Haefliger]], ''Metric spaces of non-positive curvature.'' Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319. Springer-Verlag, Berlin, 1999.</ref> motivated by ideas from Alexandrov geometry.
  <li>[http://cupuyc.free.fr/viewthread.php?tid=91181&extra=page%3D1&frombbs=1 http://cupuyc.free.fr/viewthread.php?tid=91181&extra=page%3D1&frombbs=1]</li>
*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>I. Kapovich, A. Miasnikov, P. Schupp and V. Shpilrain, ''Generic-case complexity, decision problems in group theory, and random walks.'' [[Journal of Algebra]], vol. 264 (2003), no. 2, pp. 665–694.</ref> for group-theoretic and other mathematical algorithms and algebraic rigidity results for generic groups.<ref>I. Kapovich, P. Schupp, V. Shpilrain, ''Generic properties of Whitehead's algorithm and isomorphism rigidity of random one-relator groups.'' [[Pacific Journal of Mathematics]], vol. 223 (2006), no. 1, pp. 113–140.</ref>
  <li>[http://primeraventa.com/index.php?page=item&id=155804 http://primeraventa.com/index.php?page=item&id=155804]</li>
*The study of [[automata groups]] and [[iterated monodromy group]]s as 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 spaces, particularly introduction and development of the notions of [[measure equivalence]] and [[orbit equivalence]], as well as measure-theoretic generalizations of Mostow rigidity.<ref>A. Furman, ''Gromov's measure equivalence and rigidity of higher rank lattices.'' [[Annals of Mathematics]] (2), vol. 150 (1999), no. 3, pp. 1059–1081.</ref><ref>N. Monod, Y. Shalom, ''Orbit equivalence rigidity and bounded cohomology.'' [[Annals of Mathematics]] (2), vol. 164 (2006), no. 3, pp. 825–878.</ref>
  <li>[http://neijiang.tyfo.com/news/html/?63370.html http://neijiang.tyfo.com/news/html/?63370.html]</li>
*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>M Culler and [[Karen Vogtmann|K. Vogtmann]]. ''Moduli of graphs and automorphisms of free groups.'' Inventiones Mathematicae, vol. 84 (1986), no. 1, pp. 91–119.</ref> and of the theory of [[train track (mathematics)|train tracks]]<ref>M. Bestvina and M. Handel, ''Train tracks and automorphisms of free groups.'' [[Annals of Mathematics]] (2), vol. 135 (1992), no. 1, pp. 1–51.</ref> for free group automorphisms played a particularly prominent role here.
  <li>[http://error.hostinger.eu/403.php? http://error.hostinger.eu/403.php?]</li>
*Development of [[Bass–Serre theory|Bass&ndash;Serre theory]], particularly various accessibility results<ref>M. J. Dunwoody. ''The accessibility of finitely presented groups.'' [[Inventiones Mathematicae]], vol. 81 (1985), no. 3, pp. 449–457.</ref><ref>M. Bestvina and M. Feighn. ''Bounding the complexity of simplicial group actions on trees.'' [[Inventiones Mathematicae]], vol. 103 (1991), no 3, pp. 449–469 (1991).</ref><ref>Z. Sela, ''Acylindrical accessibility for groups.'' [[Inventiones Mathematicae]], vol. 129 (1997), no. 3, pp. 527–565.</ref> and the theory of tree lattices.<ref>H. Bass and [[Alexander Lubotzky|A. Lubotzky]]. ''Tree lattices. With appendices by Bass, L. Carbone, [[Alexander Lubotzky|Lubotzky]], G. Rosenberg and J. Tits.'' Progress in Mathematics, 176. Birkhäuser Boston, Inc., Boston, MA, 2001. ISBN 0-8176-4120-3.</ref> Generalizations of Bass&ndash;Serre theory such as the theory of complexes of groups.<ref>M. R. Bridson and A. Haefliger, ''Metric spaces of non-positive curvature.'' Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319. Springer-Verlag, Berlin, 1999. ISBN 3-540-64324-9.</ref>
 
*The study of [[random walk]]s on groups and related boundary theory, particularly the notion of [[Poisson boundary]] (see, e.g.,<ref>V. A. Kaimanovich, ''The Poisson formula for groups with hyperbolic properties.'' [[Annals of Mathematics]] (2), vol. 152 (2000), no. 3, pp. 659–692.</ref>). The study of [[Amenable group|amenability]] and of groups whose amenability status is still unknown.
  <li>[http://www.kaoshi886.com/forum.php?mod=viewthread&tid=222557 http://www.kaoshi886.com/forum.php?mod=viewthread&tid=222557]</li>
*Interactions with finite group theory, particularly progress in the study of [[subgroup growth]].<ref>[[Alexander Lubotzky|A. Lubotzky]] and D. Segal. ''Subgroup growth.'' Progress in Mathematics, 212. Birkhäuser Verlag, Basel, 2003. ISBN 3-7643-6989-2.</ref>
 
*Studying subgroups and lattices in linear groups, 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 space]]s) and arithmetic methods.
</ul>
*[[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 (e.g. see <ref>M. Bestvina, M. Kapovich and B. Kleiner. ''Van Kampen's embedding obstruction for discrete groups.'' [[Inventiones Mathematicae]], vol. 150 (2002), no. 2, pp. 219–235.</ref>) homological and cohomological methods.
*Progress on traditional combinatorial group theory topics, such as the [[Burnside problem]],<ref>S. V. Ivanov. ''The free Burnside groups of sufficiently large exponents.'' International Journal of Algebra and Computation, vol. 4 (1994), no. 1–2.</ref><ref>I. G. Lysënok. ''Infinite Burnside groups of even period.'' (Russian) Izvestial Rossiyskoi Akademii Nauk Seriya Matematicheskaya, vol. 60 (1996), no. 3, pp. 3–224; translation in Izvestiya. Mathematics vol. 60 (1996), no. 3, pp. 453–654.</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 style="-moz-column-count:2; column-count:2;">
* [[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
* [[Kleinian group]]s, and other lattices acting on symmetric spaces.
* [[Wallpaper group]]s
* [[Baumslag–Solitar group]]s
* [[Graph of groups|Fundamental groups of graphs of groups]]
* [[Grigorchuk group]]
</div>
 
==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.
 
*[[B. H. Bowditch]]. ''A course on geometric group theory.'' MSJ Memoirs, 16. Mathematical Society of Japan, Tokyo, 2006. ISBN 4-931469-35-3
*M. R. Bridson and A. Haefliger, ''Metric spaces of non-positive curvature.'' Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319. Springer-Verlag, Berlin, 1999. ISBN 3-540-64324-9
* Michel Coornaert, Thomas Delzant and  Athanase Papadopoulos, "Géométrie et théorie des groupes : les groupes hyperboliques de Gromov", Lecture Notes in Mathematics, vol. 1441, Springer-Verlag, Berlin, 1990, x+165 pp.  MR 92f:57003, ISBN 3-540-52977-2
* Michel Coornaert and Athanase Papadopoulos,  Symbolic dynamics and hyperbolic groups.   Lecture Notes in Mathematics. 1539. Springer-Verlag, Berlin, 1993, viii+138 pp.  ISBN 3-540-56499-3
*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
*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. ISBN 0-86720-244-0
*M. Gromov, ''Hyperbolic Groups'', in "Essays in Group Theory" (G. M. Gersten, ed.), MSRI Publ. 8, 1987, pp.&nbsp;75–263. ISBN 0-387-96618-8
*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.&nbsp;1–295
*M. Kapovich, ''Hyperbolic manifolds and discrete groups''. Progress in Mathematics, 183. Birkhäuser Boston, Inc., Boston, MA, 2001
*[[Roger Lyndon|R. Lyndon]] and P. Schupp, ''Combinatorial Group Theory'', Springer-Verlag, Berlin, 1977. Reprinted in the "Classics in mathematics" series, 2000. ISBN 3-540-41158-5
*A. Yu. Ol'shanskii, ''Geometry of defining relations in groups.'' Translated from the 1989 Russian original by Yu. A. Bakhturin. Mathematics and its Applications (Soviet Series), 70. Kluwer Academic Publishers Group, Dordrecht, 1991
*J. Roe, ''Lectures on coarse geometry.'' University Lecture Series, 31. American Mathematical Society, Providence, RI, 2003. ISBN 0-8218-3332-4
 
==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]
*[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| ]]

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