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 can act non-trivially (that is, when the groups in question are realized as geometric symmetries or continuous transformations of some spaces).
Another important idea in geometric group theory is to consider finitely generated groups themselves as geometric objects. This is usually done by studying the Cayley 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.
In the introduction to his book Topics in Geometric Group Theory, Pierre de la Harpe wrote: "One of my personal beliefs is that fascination with symmetries and groups is one way of coping with frustrations of life's limitations: we like to recognize symmetries which allow us to recognize more than what we can see. In this sense the study of geometric group theory is a part of culture, and reminds me of several things that Georges de Rham practiced on many occasions, such as teaching mathematics, reciting Mallarmé, or greeting a friend".[1]: 3
History
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,[2] 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.
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"[8] 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,[9] 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[10][11][12] and the phrase "geometric group theory" started appearing soon afterwards. (see e.g.[13]).
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[14] 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:
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[18] and the work of Benson Farb and Lee Mosher on quasi-isometric rigidity of Baumslag–Solitar groups.[19]
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.[20] The notion of a relatively hyperbolic groups was originally introduced by Gromov in 1987[8] and refined by Farb[21] and Brian 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 famous Tarski conjectures, due to the work of Sela[23] as well as of Olga Kharlampovich and Alexei Myasnikov.[24] 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,[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 and Mark 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 or JSJ-decompositions for 3-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 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.[35][36][37]).
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.[38][39][40]
Interactions with topological dynamics in the contexts of studying actions of discrete groups on various compact spaces and group compactifications, particularly convergence group methods[42][43]
Development of the theory of group actions on -trees (particularly the Rips machine), and its applications.[44]
The study of group actions on CAT(0) spaces and CAT(0) cubical complexes,[45] motivated by ideas from Alexandrov geometry.
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 of generic-case complexity[48] for group-theoretic and other mathematical algorithms and algebraic rigidity results for generic groups.[49]
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.[52][53]
The study of Out(Fn) (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[55] and of the theory of train tracks[56] for free group automorphisms played a particularly prominent role here.
Development of Bass–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 of random walks on groups and related boundary theory, particularly the notion of Poisson boundary (see e.g.[61]). 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.[62]
Studying subgroups and lattices in linear groups, such as , 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.[63]) homological and cohomological methods.
Progress on traditional combinatorial group theory topics, such as the Burnside problem,[64][65] 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:
Fuchsian groups, Kleinian groups, and other groups acting properly discontinuously on symmetric spaces, in particular lattices in semisimple Lie 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.
^Greendlinger, Martin (1960). "Dehn's algorithm for the word problem". Communications on Pure and Applied Mathematics. 13 (1): 67–83. doi:10.1002/cpa.3160130108.
^Greendlinger, Martin (1961). "An analogue of a theorem of Magnus". Archiv der Mathematik. 12 (1): 94–96. doi:10.1007/BF01650530. S2CID120083990.
^J.-P. Serre, Trees. Translated from the 1977 French original by John Stillwell. Springer-Verlag, Berlin-New York, 1980. ISBN3-540-10103-9.
^ abMikhail Gromov, Hyperbolic Groups, in "Essays in Group Theory" (Steve M. Gersten, ed.), MSRI Publ. 8, 1987, pp. 75–263.
^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.
^Iliya Kapovich and Nadia Benakli. Boundaries of hyperbolic groups. Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001), pp. 39–93, Contemp. Math., 296, Amer. Math. Soc., Providence, RI, 2002. From the Introduction:" In the last fifteen years geometric group theory has enjoyed fast growth and rapidly increasing influence. Much of this progress has been spurred by remarkable work of M. L. Gromov [in Essays in group theory, 75–263, Springer, New York, 1987; in Geometric group theory, Vol. 2 (Sussex, 1991), 1–295, Cambridge Univ. Press, Cambridge, 1993], who has advanced the theory of word-hyperbolic groups (also referred to as Gromov-hyperbolic or negatively curved groups)."
^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]."
^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. ISBN0-521-43529-3.
^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.
^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.
^Zlil Sela, Diophantine geometry over groups and the elementary theory of free and hyperbolic groups. Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), pp. 87–92, Higher Ed. Press, Beijing, 2002.
^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.
^Rips, E.; Sela, Z. (1997). "Cyclic splittings of finitely presented groups and the canonical JSJ decomposition". Annals of Mathematics. Second Series. 146 (1): 53–109. doi:10.2307/2951832. JSTOR2951832.
^Yu, G. (1998). "The Novikov conjecture for groups with finite asymptotic dimension". Annals of Mathematics. Second Series. 147 (2): 325–355. doi:10.2307/121011. JSTOR121011.
^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.
^Marc Bourdon and Hervé Pajot. Quasi-conformal geometry and hyperbolic geometry. Rigidity in dynamics and geometry (Cambridge, 2000), pp. 1–17, Springer, Berlin, 2002.
^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.
^Y. Shalom. The algebraization of Kazhdan's property (T). International Congress of Mathematicians. Vol. II, pp. 1283–1310, Eur. Math. Soc., Zürich, 2006.
^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. ISBN0-8176-4120-3.
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. ISBN3-540-64324-9.
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. ISBN3-540-52977-2. MR1075994.
Clay, Matt; Margalit, Dan (2017). Office Hours with a Geometric Group Theorist. Princeton University Press. ISBN978-0-691-15866-2.
Coornaert, Michel; Papadopoulos, Athanase (1993). Symbolic dynamics and hyperbolic groups. Lecture Notes in Mathematics. Vol. 1539. Springer-Verlag. ISBN3-540-56499-3.
de la Harpe, P. (2000). Topics in geometric group theory. Chicago Lectures in Mathematics. University of Chicago Press. ISBN0-226-31719-6.
Gromov, M. (1987). "Hyperbolic Groups". In Gersten, G.M. (ed.). Essays in Group Theory. Vol. 8. MSRI. pp. 75–263. ISBN0-387-96618-8.
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. ISBN978-0-521-44680-8.
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