Geometric group theory
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Geometric group theory is an area in
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 became a clearly identifiable branch of mathematics in the late 1980s and early 1990s. Geometric group theory closely interacts with
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
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.[3] 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[4][5] and further developed by Roger Lyndon and Paul Schupp.[6] 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,[7] 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
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
Modern themes and developments
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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:
- The study of properties that are invariant under 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[18] and the work of Benson Farb and Lee Mosher on quasi-isometric rigidity of Baumslag–Solitar groups.[19]
- The study of properties that are invariant under
- 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 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]
- Cannon's conjecture.[41]
- 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.
- 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 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[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
- 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 unitary representations of discrete groups and Kazhdan's property (T)[54]
- The study of Out(Fn) (the 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 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 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
See also
- The ping-pong lemma, a useful way to exhibit a group as a free product
- Amenable group
- Nielsen transformation
- Tietze transformation
References
- ISBN 0-226-31721-8.
- ISBN 978-0-387-95336-6
- ^ 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.
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- ^ Roger Lyndon and Paul Schupp, Combinatorial Group Theory, Springer-Verlag, Berlin, 1977. Reprinted in the "Classics in mathematics" series, 2000.
- ISBN 3-540-10103-9.
- ^ a b Mikhail 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]."
- S2CID 120667382.
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."
- ISBN 0-521-43529-3.
- ^ 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.
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- MR 1324134.
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- ISBN 978-0-8218-1003-3.
- ^ 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.
- MR 1662319.
- ^ 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.
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- ^ 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.
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- ^ 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.
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- ^ 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.
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- ^ a b Bridson & Haefliger 1999
- ^ M. Kapovich, Hyperbolic manifolds and discrete groups. Progress in Mathematics, 183. Birkhäuser Boston, Inc., Boston, MA, 2001.
- ^ M. Gromov. Random walk in random groups. Geometric and Functional Analysis, vol. 13 (2003), no. 1, pp. 73–146.
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- ^ L. Bartholdi, R. I. Grigorchuk and Z. Sunik. Branch groups. Handbook of algebra, Vol. 3, pp. 989-1112, North-Holland, Amsterdam, 2003.
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- ^ Y. Shalom. The algebraization of Kazhdan's property (T). International Congress of Mathematicians. Vol. II, pp. 1283–1310, Eur. Math. Soc., Zürich, 2006.
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Books and monographs
These texts cover geometric group theory and related topics.
- ISBN 4-931469-35-3.
- 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. 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.
- MR 3753580.
- Epstein, D.B.A.; Cannon, J.W.; Holt, D.; Levy, S.; Paterson, M.; Thurston, W. (1992). 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.
- 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.