Geometric group theory is an area inmathematics devoted to the study offinitely generated groups via exploring the connections betweenalgebraic properties of suchgroups andtopological andgeometric properties of spaces on which these groups canact 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 theCayley graphs of groups, which, in addition to thegraph structure, are endowed with the structure of ametric space, given by the so-calledword 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 withlow-dimensional topology,hyperbolic geometry,algebraic topology,computational group theory anddifferential geometry. There are also substantial connections withcomplexity theory,mathematical logic, the study ofLie groups and their discrete subgroups,dynamical systems,probability theory,K-theory, and other areas of mathematics.

In the introduction to his bookTopics 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 thatGeorges de Rham practiced on many occasions, such as teaching mathematics, recitingMallarmé, or greeting a friend".^[1]: 3

Geometric group theory grew out ofcombinatorial group theory that largely studied properties ofdiscrete groups via analyzinggroup presentations, which describe groups asquotients offree groups; this field was first systematically studied byWalther von Dyck, student ofFelix Klein, in the early 1880s,^[2] while an early form is found in the 1856icosian calculus ofWilliam Rowan Hamilton, where he studied theicosahedral symmetry group via the edge graph of thedodecahedron. 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 ofMax Dehn,Jakob Nielsen,Kurt Reidemeister andOtto Schreier,J. H. C. Whitehead,Egbert van Kampen, amongst others, introduced some topological and geometric ideas into the study of discrete groups.^[3] Other precursors of geometric group theory includesmall cancellation theory andBass–Serre theory. Small cancellation theory was introduced byMartin Grindlinger in the 1960s^[4][5] and further developed byRoger Lyndon andPaul Schupp.^[6] It studiesvan 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,^[7] derives structural algebraic information about groups by studying group actions onsimplicial trees.External precursors of geometric group theory include the study of lattices in Lie groups, especiallyMostow's rigidity theorem, the study ofKleinian groups, and the progress achieved inlow-dimensional topology and hyperbolic geometry in the 1970s and early 1980s, spurred, in particular, byWilliam Thurston'sGeometrization 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 ofMikhail Gromov"Hyperbolic groups"^[8] that introduced the notion of ahyperbolic group (also known asword-hyperbolic orGromov-hyperbolic ornegatively curved group), which captures the idea of a finitely generated group having large-scale negative curvature, and by his subsequent monographAsymptotic Invariants of Infinite Groups,^[9] that outlined Gromov's program of understanding discrete groups up toquasi-isometry. The work of Gromov had a transformative effect on the study of discrete groups^[10][11][12] and the phrase "geometric group theory" started appearing soon afterwards. (see e.g.^[13]).

This sectionis inlist format but may read better asprose. You can help byconverting this section, if appropriate.Editing help is available.(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 isGromov's program^[14] of classifyingfinitely generated groups according to their large scale geometry. Formally, this means classifying finitely generated groups with theirword metric up toquasi-isometry. This program involves:

1. The study of properties that are invariant underquasi-isometry. Examples of such properties of finitely generated groups include: thegrowth rate of a finitely generated group; theisoperimetric function orDehn function of afinitely presented group; the number ofends of a group;hyperbolicity of a group; thehomeomorphism type of theGromov boundary of a hyperbolic group;^[15]asymptotic cones of finitely generated groups (see e.g.^[16][17]);amenability of a finitely generated group; being virtuallyabelian (that is, having an abelian subgroup of finiteindex); being virtuallynilpotent; being virtuallyfree; beingfinitely presentable; being a finitely presentable group with solvableWord Problem; and others.
2. Theorems which use quasi-isometry invariants to prove algebraic results about groups, for example:Gromov's polynomial growth theorem;Stallings' ends theorem;Mostow rigidity theorem.
3. 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 ofSchwartz on quasi-isometric rigidity of rank-one lattices^[18] and the work ofBenson Farb and Lee Mosher on quasi-isometric rigidity ofBaumslag–Solitar groups.^[19]

• The theory ofword-hyperbolic andrelatively hyperbolic groups. A particularly important development here is the work ofZlil Sela in 1990s resulting in the solution of theisomorphism problem for word-hyperbolic groups.^[20] The notion of a relatively hyperbolic groups was originally introduced by Gromov in 1987^[8] and refined by Farb^[21] andBrian Bowditch,^[22] 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 famousTarski conjectures, due to the work of Sela^[23] as well as ofOlga Kharlampovich and Alexei Myasnikov.^[24] The study oflimit groups and introduction of the language and machinery ofnon-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 ofautomatic groups,^[25] 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 andMark Sapir^[26][27] essentially characterizing the possible Dehn functions of finitely presented groups, as well as results providing explicit constructions of groups with fractional Dehn functions.^[28]
• The theory of toral orJSJ-decompositions for3-manifolds was originally brought into a group theoretic setting by Peter Kropholler.^[29] This notion has been developed by many authors for both finitely presented and finitely generated groups.^{[30][31][32][33][34]}
• Connections withgeometric analysis, the study ofC*-algebras associated with discrete groups and of the theory of free probability. This theme is represented, in particular, by considerable progress on theNovikov conjecture and theBaum–Connes conjecture and the development and study of related group-theoretic notions such as topological amenability, asymptotic dimension, uniform embeddability intoHilbert spaces, rapid decay property, and so on (see e.g.^[35][36][37]).
• Interactions with the theory of quasiconformal analysis on metric spaces, particularly in relation toCannon's conjecture about characterization of hyperbolic groups withGromov boundary homeomorphic to the 2-sphere.^[38][39][40]
• Finite subdivision rules, also in relation toCannon's conjecture.^[41]
• Interactions withtopological dynamics in the contexts of studying actions of discrete groups on various compact spaces and group compactifications, particularlyconvergence group methods^[42][43]
• Development of the theory of group actions on${\displaystyle \mathbb{R}}$-trees (particularly theRips machine), and its applications.^[44]
• The study of group actions onCAT(0) spaces and CAT(0) cubical complexes,^[45] motivated by ideas from Alexandrov geometry.
• Interactions with low-dimensional topology and hyperbolic geometry, particularly the study of 3-manifold groups (see, e.g.,^[46]),mapping class groups of surfaces,braid groups andKleinian groups.
• 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^[47] 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 ofgeneric-case complexity^[48] for group-theoretic and other mathematical algorithms and algebraic rigidity results for generic groups.^[49]
• The study ofautomata groups anditerated monodromy groups asgroups of automorphisms of infinite rooted trees. In particular,Grigorchuk's groups of intermediate growth, and their generalizations, appear in this context.^[50][51]
• The study of measure-theoretic properties of group actions onmeasure spaces, particularly introduction and development of the notions ofmeasure equivalence andorbit equivalence, as well as measure-theoretic generalizations of Mostow rigidity.^[52][53]
• The study of unitary representations of discrete groups andKazhdan's property (T)^[54]
• The study ofOut(F_n) (theouter automorphism group of afree group of rankn) and of individual automorphisms of free groups. Introduction and the study of Culler-Vogtmann'souter space^[55] and of the theory oftrain tracks^[56] for free group automorphisms played a particularly prominent role here.
• Development ofBass–Serre theory, particularly various accessibility results^[57][58][59] and the theory of tree lattices.^[60] Generalizations of Bass–Serre theory such as the theory of complexes of groups.^[45]
• The study ofrandom walks on groups and related boundary theory, particularly the notion ofPoisson boundary (see e.g.^[61]). The study ofamenability and of groups whose amenability status is still unknown.
• Interactions with finite group theory, particularly progress in the study ofsubgroup growth.^[62]
• Studying subgroups and lattices inlinear groups, such as${\displaystyle SL(n,\mathbb{R})}$, and of other Lie groups, via geometric methods (e.g.buildings),algebro-geometric tools (e.g.algebraic groups 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 withalgebraic topology and the use ofmorse-theoretic ideas in the combinatorial context; large-scale, or coarse (see e.g.^[63]) homological and cohomological methods.
• Progress on traditional combinatorial group theory topics, such as theBurnside problem,^[64][65] the study ofCoxeter groups andArtin groups, and so on (the methods used to study these questions currently are often geometric and topological).

The following examples are often studied in geometric group theory:

• Theping-pong lemma, a useful way to exhibit a group as a free product
• Amenable group
• Nielsen transformation
• Tietze transformation

These texts cover geometric group theory and related topics.

• Bowditch, Brian H. (2006).A course on geometric group theory. MSJ Memoirs. Vol. 16. Tokyo:Mathematical Society of Japan.ISBN 4-931469-35-3.
• Bridson, Martin R.;Haefliger, André (1999).Metric spaces of non-positive curvature. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Vol. 319. Berlin: Springer-Verlag.ISBN 3-540-64324-9.
• Coornaert, Michel; Delzant, Thomas; Papadopoulos, Athanase (1990).Géométrie et théorie des groupes : les groupes hyperboliques de Gromov. Lecture Notes in Mathematics. Vol. 1441. Springer-Verlag.ISBN 3-540-52977-2.MR 1075994.

• Clay, Matt; Margalit, Dan (2017).Office Hours with a Geometric Group Theorist. Princeton University Press.ISBN 978-0-691-15866-2.

• Coornaert, Michel; Papadopoulos, Athanase (1993).Symbolic dynamics and hyperbolic groups. Lecture Notes in Mathematics. Vol. 1539. Springer-Verlag.ISBN 3-540-56499-3.
• de la Harpe, P. (2000).Topics in geometric group theory. Chicago Lectures in Mathematics. University of Chicago Press.ISBN 0-226-31719-6.
• Druţu, Cornelia;Kapovich, Michael (2018).Geometric Group Theory(PDF). American Mathematical Society Colloquium Publications. Vol. 63.American Mathematical Society.ISBN 978-1-4704-1104-6.MR 3753580.
• Epstein, D.B.A.; Cannon, J.W.; Holt, D.; Levy, S.; Paterson, M.; Thurston, W. (1992).Word Processing in Groups. Jones and Bartlett.ISBN 0-86720-244-0.
• Gromov, M. (1987). "Hyperbolic Groups". In Gersten, G.M. (ed.).Essays in Group Theory. Vol. 8. MSRI. pp. 75–263.ISBN 0-387-96618-8.
• Gromov, Mikhael (1993)."Asymptotic invariants of infinite groups". In Niblo, G.A.; Roller, M.A. (eds.).Geometric Group Theory: Proceedings of the Symposium held in Sussex 1991. London Mathematical Society Lecture Note Series. Vol. 2. Cambridge University Press. pp. 1–295.ISBN 978-0-521-44680-8.
• Kapovich, M. (2001).Hyperbolic Manifolds and Discrete Groups. Progress in Mathematics. Vol. 183. Birkhäuser.ISBN 978-0-8176-3904-4.
• Lyndon, Roger C.; Schupp, Paul E. (2015) [1977].Combinatorial Group Theory. Classics in mathematics. Springer.ISBN 978-3-642-61896-3.
• Ol'shanskii, A.Yu. (2012) [1991].Geometry of Defining Relations in Groups. Springer.ISBN 978-94-011-3618-1.
• Roe, John (2003).Lectures on Coarse Geometry. University Lecture Series. Vol. 31. American Mathematical Society.ISBN 978-0-8218-3332-2.

• Jon McCammond's Geometric Group Theory Page
• What is Geometric Group Theory? By Daniel Wise
• Open Problems in combinatorial and geometric group theory
• Geometric group theory Theme on arxiv.org

• Geometric group theory
• Group theory

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