geometric group theory

Geometric group theory grew out of combinatorial group theory that largely studied properties of discrete groups via analyzing group presentations, which describe groups as quotients of free groups; this field was first systematically studied by Walther von Dyck, student of Felix Klein, in the early 1880s,{{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]

}} while an early form is found in the 1856 icosian calculus of William Rowan Hamilton, where he studied the icosahedral symmetry group via the edge graph of the dodecahedron. Currently combinatorial group theory as an area is largely subsumed by geometric group theory. Moreover, the term "geometric group theory" came to often include studying discrete groups using probabilistic, measure-theoretic, arithmetic, analytic and other approaches that lie outside of the traditional combinatorial group theory arsenal.

In the first half of the 20th century, pioneering work of Max Dehn, 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.Bruce Chandler and Wilhelm Magnus. The history of combinatorial group theory. A case study in the history of ideas. Studies in the History of Mathematics and Physical Sciences, vo. 9. Springer-Verlag, New York, 1982. Other precursors of geometric group theory include small cancellation theory and Bass–Serre theory. Small cancellation theory was introduced by Martin Grindlinger in the 1960s{{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 }}{{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 }} and further developed by Roger Lyndon and Paul Schupp.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. It studies van Kampen diagrams, corresponding to finite group presentations, via combinatorial curvature conditions and derives algebraic and algorithmic properties of groups from such analysis. Bass–Serre theory, introduced in the 1977 book of Serre,J.-P. Serre, Trees. Translated from the 1977 French original by John Stillwell. Springer-Verlag, Berlin-New York, 1980. {{ISBN|3-540-10103-9}}. derives structural algebraic information about groups by studying group actions on simplicial trees.

External precursors of geometric group theory include the study of lattices in Lie groups, especially Mostow's rigidity theorem, the study of Kleinian groups, and the progress achieved in low-dimensional topology and hyperbolic geometry in the 1970s and early 1980s, spurred, in particular, by William Thurston's Geometrization program.

The emergence of geometric group theory as a distinct area of mathematics is usually traced to the late 1980s and early 1990s. It was spurred by the 1987 monograph of Mikhail Gromov "Hyperbolic groups"Mikhail Gromov, Hyperbolic Groups, in "Essays in Group Theory" (Steve M. Gersten, ed.), MSRI Publ. 8, 1987, pp. 75–263. that introduced the notion of a hyperbolic group (also known as word-hyperbolic or Gromov-hyperbolic or negatively curved group), which captures the idea of a finitely generated group having large-scale negative curvature, and by his subsequent monograph Asymptotic Invariants of Infinite Groups,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. that outlined Gromov's program of understanding discrete groups up to quasi-isometry. The work of Gromov had a transformative effect on the study of discrete groupsIliya 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)."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]."{{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."}} and the phrase "geometric group theory" started appearing soon afterwards. (see e.g.Geometric group theory. Vol. 1. Proceedings of the symposium held at Sussex University, Sussex, July 1991. Edited by Graham A. Niblo and Martin A. Roller. London Mathematical Society Lecture Note Series, 181. Cambridge University Press, Cambridge, 1993. {{ISBN|0-521-43529-3}}.).

{{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 Gromov's programMikhail 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. of classifying finitely generated groups according to their large scale geometry. Formally, this means classifying finitely generated groups with their word metric up to 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 of a finitely generated group; the isoperimetric function or Dehn function of a finitely presented group; the number of ends of a group; hyperbolicity of a group; the homeomorphism type of the Gromov boundary of a hyperbolic group;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. asymptotic cones of finitely generated groups (see e.g.{{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 }}{{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 }}); amenability of a finitely generated group; being virtually abelian (that is, having an abelian subgroup of finite index); being virtually nilpotent; being virtually free; being finitely presentable; being a finitely presentable group with solvable Word Problem; and others.

:#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.

:#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 Schwartz on quasi-isometric rigidity of rank-one lattices{{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/ }} and the work of Benson Farb and Lee Mosher on quasi-isometric rigidity of Baumslag–Solitar groups.{{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 }}

• The theory of word-hyperbolic and relatively hyperbolic groups. A particularly important development here is the work of Zlil Sela in 1990s resulting in the solution of the isomorphism problem for word-hyperbolic groups.{{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}} The notion of a relatively hyperbolic groups was originally introduced by Gromov in 1987 and refined by Farb{{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 }} and Brian Bowditch,{{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}} 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 Tarski conjectures, due to the work of SelaZlil 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. as well as of Olga Kharlampovich and Alexei Myasnikov.{{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 }} The study of limit groups 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 groups,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. 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{{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 }}{{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 }} essentially characterizing the possible Dehn functions of finitely presented groups, as well as results providing explicit constructions of groups with fractional Dehn functions.{{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 }}
• The theory of toral or JSJ-decompositions for 3-manifolds was originally brought into a group theoretic setting by Peter Kropholler.{{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}} This notion has been developed by many authors for both finitely presented and finitely generated groups.{{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 }}{{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 }}{{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 }}{{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 }}{{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 }}
• 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 e.g.{{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 }}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.{{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 }}).
• 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.{{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 }}Marc Bourdon and Hervé Pajot. Quasi-conformal geometry and hyperbolic geometry. Rigidity in dynamics and geometry (Cambridge, 2000), pp. 1–17, Springer, Berlin, 2002.Mario Bonk, Quasiconformal geometry of fractals. International Congress of Mathematicians. Vol. II, pp. 1349–1373, Eur. Math. Soc., Zürich, 2006.
• Finite subdivision rules, also in relation to Cannon's conjecture.{{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 }}
• Interactions with topological dynamics in the contexts of studying actions of discrete groups on various compact spaces and group compactifications, particularly convergence group methodsP. 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.{{cite journal |first=Asli |last=Yaman |title=A topological characterisation of relatively hyperbolic groups |journal=Journal für die Reine und Angewandte Mathematik |volume=566 |pages=41–89 |year=2004 |mr=2039323}}
• Development of the theory of group actions on $\backslash mathbb\; R$-trees (particularly the Rips machine), and its applications.{{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 }}
• The study of group actions on CAT(0) spaces and CAT(0) cubical complexes, motivated by ideas from Alexandrov geometry.
• Interactions with low-dimensional topology and hyperbolic geometry, particularly the study of 3-manifold groups (see, e.g.,M. Kapovich, Hyperbolic manifolds and discrete groups. Progress in Mathematics, 183. Birkhäuser Boston, Inc., Boston, MA, 2001.), mapping class groups of surfaces, braid groups and Kleinian 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 proveM. Gromov. Random walk in random groups. Geometric and Functional Analysis, vol. 13 (2003), no. 1, pp. 73–146. 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{{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 }} for group-theoretic and other mathematical algorithms and algebraic rigidity results for generic groups.{{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 }}
• The study of automata groups and iterated monodromy groups as groups of automorphisms of infinite rooted trees. In particular, Grigorchuk's groups of intermediate growth, and their generalizations, appear in this context.L. Bartholdi, R. I. Grigorchuk and Z. Sunik. Branch groups. Handbook of algebra, Vol. 3, pp. 989-1112, North-Holland, Amsterdam, 2003.V. Nekrashevych. Self-similar groups. Mathematical Surveys and Monographs, 117. American Mathematical Society, Providence, RI, 2005. {{ISBN|0-8218-3831-8}}.
• 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.{{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 }}{{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 }}
• The study of unitary representations of discrete groups and Kazhdan's property (T)Y. Shalom. The algebraization of Kazhdan's property (T). International Congress of Mathematicians. Vol. II, pp. 1283–1310, Eur. Math. Soc., Zürich, 2006.
• The study of Out(F_n) (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{{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 }} and of the theory of train tracks{{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 }} for free group automorphisms played a particularly prominent role here.
• Development of Bass–Serre theory, particularly various accessibility results{{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 }}{{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 }}{{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 }} and the theory of tree lattices.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}}. Generalizations of Bass–Serre theory such as the theory of complexes of groups.{{harvnb|Bridson|Haefliger|1999}}
• The study of random walks on groups and related boundary theory, particularly the notion of Poisson boundary (see e.g.{{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 }}). The study of amenability and of groups whose amenability status is still unknown.
• Interactions with finite group theory, particularly progress in the study of subgroup growth.Alexander Lubotzky and Dan Segal. Subgroup growth. Progress in Mathematics, 212. Birkhäuser Verlag, Basel, 2003. {{ISBN|3-7643-6989-2}}. {{MR|1978431}}
• Studying subgroups and lattices in linear groups, such as $SL(n,\; \backslash mathbb\; R)$, 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 with algebraic topology and the use of morse-theoretic ideas in the combinatorial context; large-scale, or coarse (see e.g.{{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}}) homological and cohomological methods.
• Progress on traditional combinatorial group theory topics, such as the Burnside problem,{{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 }}{{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 }} the study of Coxeter groups and Artin 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:

{{div col|colwidth=25em}}

• Amenable groups
• Free Burnside groups
• The infinite cyclic group Z
• Free groups
• Free products
• Outer automorphism groups Out(F_n) (via outer space)
• Hyperbolic groups
• Mapping class groups (automorphisms of surfaces)
• Symmetric groups
• Braid groups
• Coxeter groups
• General Artin groups
• Thompson's group F
• CAT(0) groups
• Arithmetic groups
• Automatic groups
• Fuchsian groups, Kleinian groups, and other groups acting properly discontinuously on symmetric spaces, in particular lattices in semisimple Lie groups.
• Wallpaper groups
• Baumslag–Solitar groups
• Fundamental groups of graphs of groups
• Grigorchuk group

{{div col end}}

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

{{Reflist|2}}

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}}

• [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:Group theory