# Geometric group theory

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Geometric group theory

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 topological and geometric properties of spaces on which these groups act (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 graphs 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.

Geometric group theory, as a distinct area, is relatively new, and has become a clearly identifiable branch of mathematics in late 1980s and early 1990s. Geometric group theory closely interacts with low-dimensional topology, hyperbolic geometry, algebraic topology, computational group theory and geometric analysis. There are also substantial connections with complexity theory, mathematical logic, the study of Lie Groups 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 the Georges de Rham practices on many occasions, such as teaching mathematics, reciting Mallarmé, or greeting a friend" (page 3 in [ 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; 0-226-31721-8] ).

Historical background

Geometric group theory grew out of combinatorial group theory that largely studied properties of discrete groups via analyzing group presentations, that describe groups as quotients of free groups. 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 Dehn,
Nielsen, Reidemeister and
Schreier, Whitehead, 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&ndash;Serre theory.Small cancellation theory was introduced by Martin Grindlinger in 1960s [ M. Greendlinger, "Dehn's algorithm for the word problem." Communications in Pure and Applied Mathematics, vol. 13 (1960), pp. 67-83] [M. Greendlinger, "An analogue of a theorem of Magnus". Archiv der Mathematik, vol. 12 (1961), pp. 94-96 ] and further developed by Roger Lyndon and Paul Schupp [ R. Lyndon and P. 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&ndash;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 rigidity theorem, the study of Kleinian groups, and the progress achieved in low-dimensional topology and hyperbolic geometry in 1970s and early 1980s, spurred, in particular, by Thurston's 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 Gromov "Hyperbolic groups" [M. Gromov, "Hyperbolic Groups", in "Essays in Group Theory" (G. 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 Inifinite Groups" [ 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] , 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 groups [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). " ] [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] ."] [ 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." ] 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] ).

Notable themes and developments in geometric group theory

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 program [ 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] 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 boundary of a hyperbolic group [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] ; asymptotic cones of finitely generated groups (see, e.g., [T. R. Riley, "Higher connectedness of asymptotic cones." Topology, vol. 42 (2003), no. 6, pp. 1289-1352] [ L. Kramer, S. Shelah, K. Tent, and S. Thomas. "Asymptotic cones of finitely presented groups." Advances in Mathematics, vol. 193 (2005), no. 1, pp. 142-173 ] ); 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 [ R. E. Richard. "The quasi-isometry classification of rank one lattices." Institut des Hautes Études Scientifiques. Publications Mathématiques. No. 82 (1995), pp. 133-168] and the work of Farb and Mosher on quasi-isometric rigidity of Baumslag-Solitar groups. [ 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]
*The theory of word-hyperbolic and relatively hyperbolic groups. A particularly important development here is the work of Sela in 1990s resulting in the solution of the isomorphism problem for word-hyperbolic groups [Z. Sela, "The isomorphism problem for hyperbolic groups. I." Annals of Mathematics (2), vol. 141 (1995), no. 2, pp. 217-283] . The notion of a relatively hyperbolic groups was originally introduced by Gromov in 1987 [M. Gromov, "Hyperbolic Groups", in "Essays in Group Theory" (G. M. Gersten, ed.), MSRI Publ. 8, 1987, pp. 75-263] and refined by Farb [B. Farb. "Relatively hyperbolic groups." Geometric and Functional Analysis, vol. 8 (1998), no. 5, pp. 810-840] and Bowditch [ B. H. Bowditch. "Treelike structures arising from continua and convergence groups." Memoirs American Mathematical Society vol. 139 (1999), no. 662] , in 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 Tarski conjectures, due to the work of Sela [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] as well as of Kharlampovich and Myasnikov. [ 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] 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 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, Rips and Sapir [ 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
] [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
] essentially characterizing the possible Dehn functions of finitely presented groups, as well as results providing explicit constructions of groups with fractional Dehn functions. [ M. R. Bridson, "Fractional isoperimetric inequalities and subgroup distortion." Journal of the American Mathematical Society, vol. 12 (1999), no. 4, pp. 1103-1118]
*Development of the theory of JSJ-decompositions for finitely generated and finitely presented groups. [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] [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] [ 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] [B. H. Bowditch. "Cut points and canonical splittings of hyperbolic groups." Acta Mathematica, vol. 180 (1998), no. 2, pp. 145-186] [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]
*Connections with geometric analysis, the study of $\left\{mathbb C\right\}^*$-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, [ G. Yu. "The Novikov conjecture for groups with finite asymptotic dimension." Annals of Mathematics (2), vol. 147 (1998), no. 2, pp. 325-355] [ 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 ] [ I. Mineyev and G. Yu. "The Baum-Connes conjecture for hyperbolic groups." Inventiones Mathematicae, vol. 149 (2002), no. 1, pp. 97-122] ).
*Interactions with the theory of quasiconformal analysis on metric spaces, particularly in relation to Cannon's Conjecture about characterization of hyperbolic groups with boundary homeomorphic to the 2-sphere. [ M. Bonk, and B. Kleiner. "Conformal dimension and Gromov hyperbolic groups with 2-sphere boundary." Geometry and Topology, vol. 9 (2005), pp. 219-246] [M. Bourdon and H. Pajot. "Quasi-conformal geometry and hyperbolic geometry." Rigidity in dynamics and geometry (Cambridge, 2000), pp. 1-17, Springer, Berlin, 2002] [M. Bonk, "Quasiconformal geometry of fractals." International Congress of Mathematicians. Vol. II, pp. 1349-1373, Eur. Math. Soc., Zürich, 2006]
*Interactions with topological dynamics in the contexts of studying actions of discrete groups on various compact spaces and group compactifications, particularly convergence group methods [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] [ A. Yaman. "A topological charactesization of relatively hyperbolic groups." Journal für die Reine und Angewandte Mathematik, vol. 566 (2004), pp. 41-89]
*Development of the theory of group actions on $mathbb R$-trees (particularly the Rips machine), and its applications. [ M. Bestvina and M. Feighn. "Stable actions of groups on real trees." Inventiones Mathematicae, vol. 121 (1995), no. 2, pp. 287-321]
*The study of group actions on CAT(0) spaces and CAT(0) cubical complexes [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] , 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 prove [ M. 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 [ 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] for group-theoretic and other mathematical algorithms and algebraic rigidity results for generic groups. [ 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]
*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. [A. Furman, "Gromov's measure equivalence and rigidity of higher rank lattices." Annals of Mathematics (2), vol. 150 (1999), no. 3, pp. 1059-1081] [N. Monod, Y. Shalom, "Orbit equivalence rigidity and bounded cohomology." Annals of Mathematics (2), vol. 164 (2006), no. 3, pp. 825-878]
*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 [M Culler and K. Vogtmann. "Moduli of graphs and automorphisms of free groups." Inventiones Mathematicae, vol. 84 (1986), no. 1, pp. 91-119] and of the theory of train tracks [M. Bestvina and M. Handel, "Train tracks and automorphisms of free groups." Annals of Mathematics (2), vol. 135 (1992), no. 1, pp. 1-51] for free group automorphisms played a particularly prominent role here.
*Development of Bass&ndash;Serre theory, particularly various accessibility results [M. J. Dunwoody. "The accessibility of finitely presented groups." Inventiones Mathematicae, vol. 81 (1985), no. 3, pp. 449-457] [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)] [Z. Sela, "Acylindrical accessibility for groups." Inventiones Mathematicae, vol. 129 (1997), no. 3, pp. 527-565] and the theory of tree lattices. [ H. Bass, and A. Lubotzky. "Tree lattices. With appendices by Bass, L. Carbone, Lubotzky, G. Rosenberg and J. Tits." Progress in Mathematics, 176. Birkhäuser Boston, Inc., Boston, MA, 2001. ISBN 0-8176-4120-3] . Generalizations of Bass&ndash;Serre theory such as the theory of complexes of groups. [ 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 ]
*The study of random walks on groups and related boundary theory, particularly the notion of Poisson boundary (see, e.g., [V. A. Kaimanovich, "The Poisson formula for groups with hyperbolic properties." Annals of Mathematics (2), vol. 152 (2000), no. 3, pp. 659-692] ). The study of amenability and of groups whose amenability status is still unknown, such as Thompson's group "F".
*Interactions with finite group theory, particularly progress in the study of subgroup growth. [ A. Lubotzky and D. Segal. "Subgroup growth." Progress in Mathematics, 212. Birkhäuser Verlag, Basel, 2003. ISBN 3-7643-6989-2]
*Studying subgroups and lattices in linear groups, such as $SL\left(n, mathbb R\right)$, 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 (e.g. see [ 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] ) homological and cohomological methods.
*Progress on traditional combinatorial group theory topics, such as the Burnside problem [ S. V. Ivanov. "The free Burnside groups of sufficiently large exponents." International Journal of Algebra and Computation, vol. 4 (1994), no. 1-2] [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] , the study of Coxeter groups and Artin groups, 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:

* 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
* Kleinian groups, and other lattices acting on symmetric spaces.
* Wallpaper groups
* Baumslag-Solitar groups
* Fundamental groups of graphs of groups
* Grigorchuk group

References

Books and monographs on or closely related to geometric group theory

*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
*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. 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. 1-295
*M. Kapovich, "Hyperbolic manifolds and discrete groups". Progress in Mathematics, 183. Birkhäuser Boston, Inc., Boston, MA, 2001
*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

ee also

* The Cayley graph, the "canonical" choice of space for a group action
* The Ping-pong lemma, a useful way to exhibit a group as a free product
* Amenable group
* Nielsen transformation
* Tietze transformation
* Bass-Serre theory

* [http://www.math.ucsb.edu/~mccammon/geogrouptheory/ John 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]

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