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Geometric group theory
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<h2 id="history">History</h2> <p>Geometric group theory grew out of <a href="/facts/Combinatorial_group_theory/dh5VHLga">combinatorial group theory</a> that largely studied properties of <a href="/facts/Discrete_group/HIjkpJhS">discrete groups</a> via analyzing <a href="/facts/Presentation_of_a_group/f9emPUoI">group presentations</a>, which describe groups as <a href="/facts/Quotient_group/HlwpHtjb">quotients</a> of <a href="/facts/Free_group/zPo2NIGb">free groups</a>; this field was first systematically studied by <a href="/facts/Walther_von_Dyck/lXuECVC4">Walther von Dyck</a>, student of <a href="/facts/Felix_Klein/XMILhEfy">Felix Klein</a>, in the early 1880s,<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> while an early form is found in the 1856 <a href="/facts/Icosian_calculus/ptfEWNnG">icosian calculus</a> of <a href="/facts/William_Rowan_Hamilton/U86CqUlt">William Rowan Hamilton</a>, where he studied the <a href="/facts/Icosahedral_symmetry/pDlwAy23">icosahedral symmetry</a> group via the edge graph of the <a href="/facts/Dodecahedron/7r519eCl">dodecahedron</a>. 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, <a href="/facts/Measure_theory/yCq7zlI4">measure-theoretic</a>, arithmetic, analytic and other approaches that lie outside of the traditional combinatorial group theory arsenal. </p><p>In the first half of the 20th century, pioneering work of <a href="/facts/Max_Dehn/4x7x0CM0">Max Dehn</a>, <a href="/facts/Jakob_Nielsen_(mathematician)/kkAvFr7f">Jakob Nielsen</a>, <a href="/facts/Kurt_Reidemeister/YYPW5sJ8">Kurt Reidemeister</a> and <a href="/facts/Otto_Schreier/rbq7ng4v">Otto Schreier</a>, <a href="/facts/J._H._C._Whitehead/NhjAGNjR">J. H. C. Whitehead</a>, <a href="/facts/Egbert_van_Kampen/ItPXz3vB">Egbert van Kampen</a>, amongst others, introduced some topological and geometric ideas into the study of discrete groups.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> Other precursors of geometric group theory include <a href="/facts/Small_cancellation_theory/UVJBUY2z">small cancellation theory</a> and <a href="/facts/Bass%25E2%2580%2593Serre_theory/qCI1VD4g">Bass–Serre theory</a>. Small cancellation theory was introduced by Martin Grindlinger in the 1960s<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a><a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> and further developed by <a href="/facts/Roger_Lyndon/ESCf6Rr3">Roger Lyndon</a> and <a href="/facts/Paul_Schupp/HLUrqI6b">Paul Schupp</a>.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> It studies <a href="/facts/Van_Kampen_diagram/GJfBACpV">van Kampen diagrams</a>, 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,<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> derives structural algebraic information about groups by studying group actions on <a href="/facts/Tree_(graph_theory)/TCWk4Joy">simplicial trees</a>. External precursors of geometric group theory include the study of lattices in Lie groups, especially <a href="/facts/Mostow%2527s_rigidity_theorem/KXtuIhnr">Mostow's rigidity theorem</a>, the study of <a href="/facts/Kleinian_group/FBMCx9Ry">Kleinian groups</a>, and the progress achieved in <a href="/facts/Low-dimensional_topology/o5g9aRGw">low-dimensional topology</a> and hyperbolic geometry in the 1970s and early 1980s, spurred, in particular, by <a href="/facts/William_Thurston/K1Qe8801">William Thurston</a>'s <a href="/facts/Geometrization_conjecture/FDpqIM7y">Geometrization program</a>. </p><p>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 <a href="/facts/Mikhail_Gromov_(mathematician)/N0t5hhcI">Mikhail Gromov</a> <i>"Hyperbolic groups"</i><a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> that introduced the notion of a <a href="/facts/Hyperbolic_group/yXGUebC4">hyperbolic group</a> (also known as <i>word-hyperbolic</i> or <i>Gromov-hyperbolic</i> or <i>negatively curved</i> group), which captures the idea of a finitely generated group having large-scale negative curvature, and by his subsequent monograph <i>Asymptotic Invariants of Infinite Groups</i>,<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> that outlined Gromov's program of understanding discrete groups up to <a href="/facts/Glossary_of_Riemannian_and_metric_geometry/e0aAePjd">quasi-isometry</a>. The work of Gromov had a transformative effect on the study of discrete groups<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a><a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a><a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> and the phrase "geometric group theory" started appearing soon afterwards. (see e.g.<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a>). </p> <h2 id="modern-themes-and-developments">Modern themes and developments</h2> <p>Notable themes and developments in geometric group theory in 1990s and 2000s include: </p> <ul><li>Gromov's program to study quasi-isometric properties of groups.</li></ul> A particularly influential broad theme in the area is <a href="/facts/Mikhail_Gromov_(mathematician)/N0t5hhcI">Gromov</a>'s program<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> of classifying <a href="/facts/Generating_set_of_a_group/B9q5lnFY">finitely generated groups</a> according to their large scale geometry. Formally, this means classifying finitely generated groups with their <a href="/facts/Word_metric/NMqoPzyx">word metric</a> up to <a href="/facts/Glossary_of_Riemannian_and_metric_geometry/e0aAePjd">quasi-isometry</a>. This program involves: <ol><li>The study of properties that are invariant under <a href="/facts/Quasi-isometry/0n6JXPQg">quasi-isometry</a>. Examples of such properties of finitely generated groups include: the <a href="/facts/Growth_rate_(group_theory)/IlKFW6Fj">growth rate</a> of a finitely generated group; the <a href="/facts/Dehn_function/5XGpG0I9">isoperimetric function</a> or <a href="/facts/Van_Kampen_diagram/GJfBACpV">Dehn function</a> of a <a href="/facts/Finitely_presented_group/f9emPUoI">finitely presented group</a>; the number of <a href="/facts/End_(topology)/Rtwywetf">ends of a group</a>; <a href="/facts/Hyperbolic_group/yXGUebC4">hyperbolicity of a group</a>; the <a href="/facts/Homeomorphism/jMfWg3Mn">homeomorphism</a> type of the <a href="/facts/Gromov_boundary/ySBT5k8J">Gromov boundary</a> of a hyperbolic group;<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a> <a href="/facts/Ultralimit/aC31JMos">asymptotic cones</a> of finitely generated groups (see e.g.<a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a><a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a>); <a href="/facts/Amenable_group/08XrAwvx">amenability</a> of a finitely generated group; being virtually <a href="/facts/Abelian_group/FfWwmx4R">abelian</a> (that is, having an abelian subgroup of finite <a href="/facts/Index_of_a_subgroup/ewY2A71Y">index</a>); being virtually <a href="/facts/Nilpotent_group/BJuochky">nilpotent</a>; being virtually <a href="/facts/Free_group/zPo2NIGb">free</a>; being <a href="/facts/Finitely_presented_group/f9emPUoI">finitely presentable</a>; being a finitely presentable group with solvable <a href="/facts/Word_problem_for_groups/eKBrkBPv">Word Problem</a>; and others.</li> <li>Theorems which use quasi-isometry invariants to prove algebraic results about groups, for example: <a href="/facts/Gromov%2527s_theorem_on_groups_of_polynomial_growth/RI8qoSuJ">Gromov's polynomial growth theorem</a>; <a href="/facts/Stallings_theorem_about_ends_of_groups/xB6QPhuB">Stallings' ends theorem</a>; <a href="/facts/Mostow_rigidity_theorem/KXtuIhnr">Mostow rigidity theorem</a>.</li> <li>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 <a href="/facts/Richard_Schwartz_(mathematician)/vwbyqYqw">Schwartz</a> on quasi-isometric rigidity of rank-one lattices<a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a> and the work of <a href="/facts/Benson_Farb/vkCSH1ad">Benson Farb</a> and Lee Mosher on quasi-isometric rigidity of <a href="/facts/Baumslag%25E2%2580%2593Solitar_group/IThTL9oM">Baumslag–Solitar groups</a>.<a class="footnote-ref" id="fnref:19" href="#fn:19"><sup>19</sup></a></li></ol> <ul><li>The theory of <a href="/facts/Hyperbolic_group/yXGUebC4">word-hyperbolic</a> and <a href="/facts/Relatively_hyperbolic_group/u7336Qba">relatively hyperbolic</a> groups. A particularly important development here is the work of <a href="/facts/Zlil_Sela/xKBSKGDl">Zlil Sela</a> in 1990s resulting in the solution of the <a href="/facts/Group_isomorphism_problem/FEWC7TC7">isomorphism problem</a> for word-hyperbolic groups.<a class="footnote-ref" id="fnref:20" href="#fn:20"><sup>20</sup></a> The notion of a relatively hyperbolic groups was originally introduced by Gromov in 1987<a class="footnote-ref" id="fnref:21" href="#fn:21"><sup>21</sup></a> and refined by Farb<a class="footnote-ref" id="fnref:22" href="#fn:22"><sup>22</sup></a> and <a href="/facts/Brian_Bowditch/0BMSFmTp">Brian Bowditch</a>,<a class="footnote-ref" id="fnref:23" href="#fn:23"><sup>23</sup></a> in the 1990s. The study of relatively hyperbolic groups gained prominence in the 2000s.</li> <li>Interactions with mathematical logic and the study of the first-order theory of free groups. Particularly important progress occurred on the famous <a href="/facts/Free_group/zPo2NIGb">Tarski conjectures</a>, due to the work of Sela<a class="footnote-ref" id="fnref:24" href="#fn:24"><sup>24</sup></a> as well as of <a href="/facts/Olga_Kharlampovich/QvuVg0S7">Olga Kharlampovich</a> and Alexei Myasnikov.<a class="footnote-ref" id="fnref:25" href="#fn:25"><sup>25</sup></a> The study of limit groups and introduction of the language and machinery of <a href="/facts/Noncommutative_algebraic_geometry/D03KU3Gp">non-commutative algebraic geometry</a> gained prominence.</li> <li>Interactions with computer science, complexity theory and the theory of formal languages. This theme is exemplified by the development of the theory of <a href="/facts/Automatic_group/HN1Lj7TG">automatic groups</a>,<a class="footnote-ref" id="fnref:26" href="#fn:26"><sup>26</sup></a> a notion that imposes certain geometric and language theoretic conditions on the multiplication operation in a finitely generated group.</li> <li>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, <a href="/facts/Eliyahu_Rips/z1ASKrso">Eliyahu Rips</a> and <a href="/facts/Mark_Sapir/bjzSH5S9">Mark Sapir</a><a class="footnote-ref" id="fnref:27" href="#fn:27"><sup>27</sup></a><a class="footnote-ref" id="fnref:28" href="#fn:28"><sup>28</sup></a> essentially characterizing the possible Dehn functions of finitely presented groups, as well as results providing explicit constructions of groups with fractional Dehn functions.<a class="footnote-ref" id="fnref:29" href="#fn:29"><sup>29</sup></a></li> <li>The theory of toral or <a href="/facts/JSJ_decomposition/6y5kyIya">JSJ-decompositions</a> for <a href="/facts/3-manifold/48yr0Tzx">3-manifolds</a> was originally brought into a group theoretic setting by Peter Kropholler.<a class="footnote-ref" id="fnref:30" href="#fn:30"><sup>30</sup></a> This notion has been developed by many authors for both finitely presented and finitely generated groups.<a class="footnote-ref" id="fnref:31" href="#fn:31"><sup>31</sup></a><a class="footnote-ref" id="fnref:32" href="#fn:32"><sup>32</sup></a><a class="footnote-ref" id="fnref:33" href="#fn:33"><sup>33</sup></a><a class="footnote-ref" id="fnref:34" href="#fn:34"><sup>34</sup></a><a class="footnote-ref" id="fnref:35" href="#fn:35"><sup>35</sup></a></li> <li>Connections with <a href="/facts/Geometric_analysis/wLtqsMAe">geometric analysis</a>, the study of <a href="/facts/C*-algebras/B3tnGHIo">C*-algebras</a> associated with discrete groups and of the theory of free probability. This theme is represented, in particular, by considerable progress on the <a href="/facts/Novikov_conjecture/ADvc1bUD">Novikov conjecture</a> and the <a href="/facts/Baum%25E2%2580%2593Connes_conjecture/YHRAAG7z">Baum–Connes conjecture</a> and the development and study of related group-theoretic notions such as topological amenability, asymptotic dimension, uniform embeddability into <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert spaces</a>, rapid decay property, and so on (see e.g.<a class="footnote-ref" id="fnref:36" href="#fn:36"><sup>36</sup></a><a class="footnote-ref" id="fnref:37" href="#fn:37"><sup>37</sup></a><a class="footnote-ref" id="fnref:38" href="#fn:38"><sup>38</sup></a>).</li> <li>Interactions with the theory of quasiconformal analysis on metric spaces, particularly in relation to <a href="/facts/Cannon%2527s_conjecture/M40dDJUm">Cannon's conjecture</a> about characterization of hyperbolic groups with <a href="/facts/Gromov_boundary/ySBT5k8J">Gromov boundary</a> homeomorphic to the 2-sphere.<a class="footnote-ref" id="fnref:39" href="#fn:39"><sup>39</sup></a><a class="footnote-ref" id="fnref:40" href="#fn:40"><sup>40</sup></a><a class="footnote-ref" id="fnref:41" href="#fn:41"><sup>41</sup></a></li> <li><a href="/facts/Finite_subdivision_rules/M40dDJUm">Finite subdivision rules</a>, also in relation to <a href="/facts/Cannon%2527s_conjecture/M40dDJUm">Cannon's conjecture</a>.<a class="footnote-ref" id="fnref:42" href="#fn:42"><sup>42</sup></a></li> <li>Interactions with <a href="/facts/Topological_dynamics/QYCym7RF">topological dynamics</a> in the contexts of studying actions of discrete groups on various compact spaces and group compactifications, particularly <a href="/facts/Convergence_group/0eTK9b3V">convergence group</a> methods<a class="footnote-ref" id="fnref:43" href="#fn:43"><sup>43</sup></a><a class="footnote-ref" id="fnref:44" href="#fn:44"><sup>44</sup></a></li> <li>Development of the theory of group actions on <a href="/facts/Real_tree/KUBIkuLE"> R {\displaystyle \mathbb {R} } -trees</a> (particularly the <a href="/facts/Rips_machine/FeUJLVlo">Rips machine</a>), and its applications.<a class="footnote-ref" id="fnref:45" href="#fn:45"><sup>45</sup></a></li> <li>The study of group actions on <a href="/facts/CAT(0)_space/fmnJjYsN">CAT(0) spaces</a> and CAT(0) cubical complexes,<a class="footnote-ref" id="fnref:46" href="#fn:46"><sup>46</sup></a> motivated by ideas from Alexandrov geometry.</li> <li>Interactions with low-dimensional topology and hyperbolic geometry, particularly the study of 3-manifold groups (see, e.g.,<a class="footnote-ref" id="fnref:47" href="#fn:47"><sup>47</sup></a>), <a href="/facts/Mapping_class_group/buQm001q">mapping class groups</a> of surfaces, <a href="/facts/Braid_group/rn5323Kx">braid groups</a> and <a href="/facts/Kleinian_group/FBMCx9Ry">Kleinian groups</a>.</li> <li>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<a class="footnote-ref" id="fnref:48" href="#fn:48"><sup>48</sup></a> 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 <a href="/facts/Generic-case_complexity/StY0bfN3">generic-case complexity</a><a class="footnote-ref" id="fnref:49" href="#fn:49"><sup>49</sup></a> for group-theoretic and other mathematical algorithms and algebraic rigidity results for generic groups.<a class="footnote-ref" id="fnref:50" href="#fn:50"><sup>50</sup></a></li> <li>The study of automata groups and <a href="/facts/Iterated_monodromy_group/nRujhhcT">iterated monodromy groups</a> as <a href="/facts/Automorphism_group/65vImKwe">groups of automorphisms</a> of infinite rooted trees. In particular, <a href="/facts/Grigorchuk%2527s_group/HahdoDWp">Grigorchuk's groups</a> of intermediate growth, and their generalizations, appear in this context.<a class="footnote-ref" id="fnref:51" href="#fn:51"><sup>51</sup></a><a class="footnote-ref" id="fnref:52" href="#fn:52"><sup>52</sup></a></li> <li>The study of measure-theoretic properties of group actions on <a href="/facts/Measure_space/w9Dx4KY1">measure spaces</a>, particularly introduction and development of the notions of measure equivalence and orbit equivalence, as well as measure-theoretic generalizations of Mostow rigidity.<a class="footnote-ref" id="fnref:53" href="#fn:53"><sup>53</sup></a><a class="footnote-ref" id="fnref:54" href="#fn:54"><sup>54</sup></a></li> <li>The study of unitary representations of discrete groups and <a href="/facts/Kazhdan%2527s_property_(T)/v7D8F9sW">Kazhdan's property (T)</a><a class="footnote-ref" id="fnref:55" href="#fn:55"><sup>55</sup></a></li> <li>The study of <i>Out</i>(<i>F</i><i>n</i>) (the <a href="/facts/Outer_automorphism_group/wb5qwIxb">outer automorphism group</a> of a <a href="/facts/Free_group/zPo2NIGb">free group</a> of rank <i>n</i>) and of individual automorphisms of free groups. Introduction and the study of Culler-Vogtmann's <a href="/facts/Outer_space_(group_theory)/G5cCvetC">outer space</a><a class="footnote-ref" id="fnref:56" href="#fn:56"><sup>56</sup></a> and of the theory of <a href="/facts/Train_track_(mathematics)/ac485Dwf">train tracks</a><a class="footnote-ref" id="fnref:57" href="#fn:57"><sup>57</sup></a> for free group automorphisms played a particularly prominent role here.</li> <li>Development of <a href="/facts/Bass%25E2%2580%2593Serre_theory/qCI1VD4g">Bass–Serre theory</a>, particularly various accessibility results<a class="footnote-ref" id="fnref:58" href="#fn:58"><sup>58</sup></a><a class="footnote-ref" id="fnref:59" href="#fn:59"><sup>59</sup></a><a class="footnote-ref" id="fnref:60" href="#fn:60"><sup>60</sup></a> and the theory of tree lattices.<a class="footnote-ref" id="fnref:61" href="#fn:61"><sup>61</sup></a> Generalizations of Bass–Serre theory such as the theory of complexes of groups.<a class="footnote-ref" id="fnref:62" href="#fn:62"><sup>62</sup></a></li> <li>The study of <a href="/facts/Random_walk/08v0jmfv">random walks</a> on groups and related boundary theory, particularly the notion of <a href="/facts/Poisson_boundary/1KvVw6mG">Poisson boundary</a> (see e.g.<a class="footnote-ref" id="fnref:63" href="#fn:63"><sup>63</sup></a>). The study of <a href="/facts/Amenable_group/08XrAwvx">amenability</a> and of groups whose amenability status is still unknown.</li> <li>Interactions with finite group theory, particularly progress in the study of <a href="/facts/Subgroup_growth/e7Sk5FNn">subgroup growth</a>.<a class="footnote-ref" id="fnref:64" href="#fn:64"><sup>64</sup></a></li> <li>Studying subgroups and lattices in <a href="/facts/Linear_group/wVzG0sa3">linear groups</a>, such as S L ( n , R ) {\displaystyle SL(n,\mathbb {R} )} , and of other Lie groups, via geometric methods (e.g. <a href="/facts/Building_(mathematics)/916RYbEN">buildings</a>), <a href="/facts/Algebraic_geometry/rh7RHVp3">algebro-geometric</a> tools (e.g. <a href="/facts/Algebraic_group/DDtBCIvh">algebraic groups</a> and representation varieties), analytic methods (e.g. unitary representations on Hilbert spaces) and arithmetic methods.</li> <li><a href="/facts/Group_cohomology/ncLTIHj4">Group cohomology</a>, using algebraic and topological methods, particularly involving interaction with <a href="/facts/Algebraic_topology/2wA8yrjT">algebraic topology</a> and the use of <a href="/facts/Morse_theory/bJRqH8BQ">morse-theoretic</a> ideas in the combinatorial context; large-scale, or coarse (see e.g.<a class="footnote-ref" id="fnref:65" href="#fn:65"><sup>65</sup></a>) homological and cohomological methods.</li> <li>Progress on traditional combinatorial group theory topics, such as the <a href="/facts/Burnside_problem/RsWB8d31">Burnside problem</a>,<a class="footnote-ref" id="fnref:66" href="#fn:66"><sup>66</sup></a><a class="footnote-ref" id="fnref:67" href="#fn:67"><sup>67</sup></a> the study of <a href="/facts/Coxeter_group/bgzHFm1Q">Coxeter groups</a> and <a href="/facts/Artin_group/9BQbTGE8">Artin groups</a>, and so on (the methods used to study these questions currently are often geometric and topological).</li></ul> <h2 id="examples">Examples</h2> <p>The following examples are often studied in geometric group theory: </p> <ul><li><a href="/facts/Amenable_group/08XrAwvx">Amenable groups</a></li> <li><a href="/facts/Burnside_group/RsWB8d31">Free Burnside groups</a></li> <li>The infinite <a href="/facts/Cyclic_group/JL5zfFuL">cyclic group</a> <a href="/facts/Integer/PUwwrawx">Z</a></li> <li><a href="/facts/Free_group/zPo2NIGb">Free groups</a></li> <li><a href="/facts/Free_product/bJsMaMGD">Free products</a></li> <li><a href="/facts/Outer_automorphism_group/wb5qwIxb">Outer automorphism groups</a> <a href="/facts/Out(Fn)/G5cCvetC">Out(F<i>n</i>)</a> (via <a href="/facts/Outer_space_(group_theory)/G5cCvetC">outer space</a>)</li> <li><a href="/facts/Hyperbolic_group/yXGUebC4">Hyperbolic groups</a></li> <li><a href="/facts/Mapping_class_group/buQm001q">Mapping class groups</a> (automorphisms of surfaces)</li> <li><a href="/facts/Symmetric_group/RUksK5l5">Symmetric groups</a></li> <li><a href="/facts/Braid_group/rn5323Kx">Braid groups</a></li> <li><a href="/facts/Coxeter_group/bgzHFm1Q">Coxeter groups</a></li> <li>General <a href="/facts/Artin_group/9BQbTGE8">Artin groups</a></li> <li><a href="/facts/Thompson_groups/gGa03pvd">Thompson's group</a> <i>F</i></li> <li><a href="/facts/CAT(0)_group/aEJDKhEr">CAT(0) groups</a></li> <li><a href="/facts/Arithmetic_group/I8eA2483">Arithmetic groups</a></li> <li><a href="/facts/Automatic_group/HN1Lj7TG">Automatic groups</a></li> <li><a href="/facts/Fuchsian_group/qNrHQeUq">Fuchsian groups</a>, <a href="/facts/Kleinian_group/FBMCx9Ry">Kleinian groups</a>, and other groups acting properly discontinuously on symmetric spaces, in particular <a href="/facts/Lattice_(discrete_subgroup)/fLr9dYhz">lattices</a> in semisimple Lie groups.</li> <li><a href="/facts/Wallpaper_group/egxEaC8q">Wallpaper groups</a></li> <li><a href="/facts/Baumslag%25E2%2580%2593Solitar_group/IThTL9oM">Baumslag–Solitar groups</a></li> <li><a href="/facts/Graph_of_groups/4QdPyWxB">Fundamental groups of graphs of groups</a></li> <li><a href="/facts/Grigorchuk_group/HahdoDWp">Grigorchuk group</a></li></ul> <h2 id="see-also">See also</h2> <ul><li>The <a href="/facts/Ping-pong_lemma/5zojWwQj">ping-pong lemma</a>, a useful way to exhibit a group as a free product</li> <li><a href="/facts/Amenable_group/08XrAwvx">Amenable group</a></li> <li><a href="/facts/Nielsen_transformation/tx5VeX06">Nielsen transformation</a></li> <li><a href="/facts/Tietze_transformation/whL6MwF1">Tietze transformation</a></li></ul> <h3>Books and monographs</h3> <p>These texts cover geometric group theory and related topics. </p> <ul><li><a href="/facts/Brian_Bowditch/0BMSFmTp">Bowditch, Brian H.</a> (2006). <i>A course on geometric group theory</i>. MSJ Memoirs. Vol. 16. Tokyo: <a href="/facts/Mathematical_Society_of_Japan/7sAAMXe2">Mathematical Society of Japan</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 4-931469-35-3.</li> <li><a href="/facts/Martin_Bridson/q63MNtcW">Bridson, Martin R.</a>; <a href="/facts/Andr%25C3%25A9_Haefliger/VFEy0jaS">Haefliger, André</a> (1999). <i>Metric spaces of non-positive curvature</i>. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Vol. 319. Berlin: Springer-Verlag. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-64324-9.</li> <li>Coornaert, Michel; Delzant, Thomas; Papadopoulos, Athanase (1990). <i>Géométrie et théorie des groupes : les groupes hyperboliques de Gromov</i>. Lecture Notes in Mathematics. Vol. 1441. Springer-Verlag. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-52977-2. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1075994">1075994</a>.</li></ul> <ul><li>Clay, Matt; Margalit, Dan (2017). <i>Office Hours with a Geometric Group Theorist</i>. Princeton University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-691-15866-2.</li></ul> <ul><li>Coornaert, Michel; Papadopoulos, Athanase (1993). <i>Symbolic dynamics and hyperbolic groups</i>. Lecture Notes in Mathematics. Vol. 1539. Springer-Verlag. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-56499-3.</li> <li>de la Harpe, P. (2000). <i>Topics in geometric group theory</i>. Chicago Lectures in Mathematics. University of Chicago Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-226-31719-6.</li> <li><a href="/facts/Cornelia_Dru%25C8%259Bu/lLlz6WJ2">Druţu, Cornelia</a>; <a href="/facts/Michael_Kapovich/dNoJHzkH">Kapovich, Michael</a> (2018). <a href="https://www.math.ucdavis.edu/~kapovich/EPR/ggt.pdf"><i>Geometric Group Theory</i></a> (PDF). American Mathematical Society Colloquium Publications. Vol. 63. <a href="/facts/American_Mathematical_Society/fr5gotCt">American Mathematical Society</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4704-1104-6. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=3753580">3753580</a>.</li> <li>Epstein, D.B.A.; Cannon, J.W.; Holt, D.; Levy, S.; Paterson, M.; Thurston, W. (1992). <a href="/facts/Word_Processing_in_Groups/rfHNNMIS"><i>Word Processing in Groups</i></a>. Jones and Bartlett. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-86720-244-0.</li> <li>Gromov, M. (1987). "Hyperbolic Groups". In Gersten, G.M. (ed.). <i>Essays in Group Theory</i>. Vol. 8. MSRI. pp. 75–263. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-96618-8.</li> <li>Gromov, Mikhael (1993). <a href="https://books.google.com/books?id=dH02YAfVqkYC&pg=PP1">"Asymptotic invariants of infinite groups"</a>. In Niblo, G.A.; Roller, M.A. (eds.). <i>Geometric Group Theory: Proceedings of the Symposium held in Sussex 1991</i>. London Mathematical Society Lecture Note Series. Vol. 2. Cambridge University Press. pp. 1–295. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-44680-8.</li> <li>Kapovich, M. (2001). <a href="https://books.google.com/books?id=YmphheDo18kC"><i>Hyperbolic Manifolds and Discrete Groups</i></a>. Progress in Mathematics. Vol. 183. Birkhäuser. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8176-3904-4.</li> <li><a href="/facts/Roger_Lyndon/ESCf6Rr3">Lyndon, Roger C.</a>; Schupp, Paul E. (2015) [1977]. <a href="https://books.google.com/books?id=cOLrCAAAQBAJ"><i>Combinatorial Group Theory</i></a>. Classics in mathematics. Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-642-61896-3.</li> <li>Ol'shanskii, A.Yu. (2012) [1991]. <a href="https://books.google.com/books?id=uS_pCAAAQBAJ&pg=PP1"><i>Geometry of Defining Relations in Groups</i></a>. Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-94-011-3618-1.</li> <li>Roe, John (2003). <a href="https://books.google.com/books?id=jbsFCAAAQBAJ"><i>Lectures on Coarse Geometry</i></a>. University Lecture Series. Vol. 31. American Mathematical Society. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8218-3332-2.</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="http://www.math.ucsb.edu/~mccammon/geogrouptheory/">Jon McCammond's Geometric Group Theory Page</a></li> <li><a href="http://www.math.mcgill.ca/wise/ggt/cayley.html"><i>What is Geometric Group Theory?</i> By Daniel Wise</a></li> <li><a href="https://web.archive.org/web/20040830075241/http://zebra.sci.ccny.cuny.edu/web/nygtc/problems/">Open Problems in combinatorial and geometric group theory</a></li> <li><a href="http://xstructure.inr.ac.ru/x-bin/theme3.py?level=1&index1=-98867">Geometric group theory Theme on arxiv.org</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>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, ISBN 0-226-31721-8. <a href="https://books.google.com/books?id=60fTzwfqeQIC&dq=de+la+Harpe,+Topics+in+geometric+group+theory&pg=PP1" target="_blank">https://books.google.com/books?id=60fTzwfqeQIC&dq=de+la+Harpe,+Topics+in+geometric+group+theory&pg=PP1</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Stillwell, John (2002), Mathematics and its history, Springer, p. 374, ISBN 978-0-387-95336-6 <a href="978-0-387-95336-6" target="_blank">978-0-387-95336-6</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>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. <a href="/wiki/Wilhelm_Magnus" target="_blank">/wiki/Wilhelm_Magnus</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>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. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Greendlinger, Martin (1961). "An analogue of a theorem of Magnus". Archiv der Mathematik. 12 (1): 94–96. doi:10.1007/BF01650530. S2CID 120083990. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Roger Lyndon and Paul Schupp, Combinatorial Group Theory, Springer-Verlag, Berlin, 1977. Reprinted in the "Classics in mathematics" series, 2000. <a href="/wiki/Roger_Lyndon" target="_blank">/wiki/Roger_Lyndon</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>J.-P. Serre, Trees. Translated from the 1977 French original by John Stillwell. Springer-Verlag, Berlin-New York, 1980. ISBN 3-540-10103-9. <a href="/wiki/John_Stillwell" target="_blank">/wiki/John_Stillwell</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Mikhail Gromov, Hyperbolic Groups, in "Essays in Group Theory" (Steve M. Gersten, ed.), MSRI Publ. 8, 1987, pp. 75–263. <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>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. <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>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)." <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>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]." <a href="/wiki/Brian_Bowditch" target="_blank">/wiki/Brian_Bowditch</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Elek, Gabor (2006). "The mathematics of Misha Gromov". Acta Mathematica Hungarica. 113 (3): 171–185. doi:10.1007/s10474-006-0098-5. 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." <a href="https://doi.org/10.1007%2Fs10474-006-0098-5" target="_blank">https://doi.org/10.1007%2Fs10474-006-0098-5</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>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. <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>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. <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>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. <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> <li id="fn:16"><p>Riley, Tim R. (2003). "Higher connectedness of asymptotic cones". Topology. 42 (6): 1289–1352. doi:10.1016/S0040-9383(03)00002-8. <a href="https://doi.org/10.1016%2FS0040-9383%2803%2900002-8" target="_blank">https://doi.org/10.1016%2FS0040-9383%2803%2900002-8</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></p></li> <li id="fn:17"><p>Kramer, Linus; Shelah, Saharon; Tent, Katrin; Thomas, Simon (2005). "Asymptotic cones of finitely presented groups". Advances in Mathematics. 193 (1): 142–173. arXiv:math/0306420. doi:10.1016/j.aim.2004.04.012. S2CID 4769970. <a href="/wiki/Saharon_Shelah" target="_blank">/wiki/Saharon_Shelah</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></p></li> <li id="fn:18"><p>Schwartz, R.E. (1995). "The quasi-isometry classification of rank one lattices". Publications Mathématiques de l'Institut des Hautes Études Scientifiques. 82 (1): 133–168. doi:10.1007/BF02698639. S2CID 67824718. <a href="http://www.numdam.org/item/PMIHES_1995__82__133_0/" target="_blank">http://www.numdam.org/item/PMIHES_1995__82__133_0/</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></p></li> <li id="fn:19"><p>Farb, Benson; Mosher, Lee (1998). "A rigidity theorem for the solvable Baumslag–Solitar groups. With an appendix by Daryl Cooper". Inventiones Mathematicae. 131 (2): 419–451. doi:10.1007/s002220050210. MR 1608595. S2CID 121180189. <a href="/wiki/Benson_Farb" target="_blank">/wiki/Benson_Farb</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></p></li> <li id="fn:20"><p>Sela, Zlil (1995). "The isomorphism problem for hyperbolic groups. I". Annals of Mathematics. (2). 141 (2): 217–283. doi:10.2307/2118520. JSTOR 2118520. MR 1324134. <a href="/wiki/Annals_of_Mathematics" target="_blank">/wiki/Annals_of_Mathematics</a> <a href="#fnref:20" class="footnote-back-ref">↩</a></p></li> <li id="fn:21"><p>Mikhail Gromov, Hyperbolic Groups, in "Essays in Group Theory" (Steve M. Gersten, ed.), MSRI Publ. 8, 1987, pp. 75–263. <a href="#fnref:21" class="footnote-back-ref">↩</a></p></li> <li id="fn:22"><p>Farb, Benson (1998). "Relatively hyperbolic groups". Geometric and Functional Analysis. 8 (5): 810–840. doi:10.1007/s000390050075. MR 1650094. S2CID 123370926. <a href="/wiki/Benson_Farb" target="_blank">/wiki/Benson_Farb</a> <a href="#fnref:22" class="footnote-back-ref">↩</a></p></li> <li id="fn:23"><p>Bowditch, Brian H. (1999). Treelike Structures Arising from Continua and Convergence Groups. Memoirs American Mathematical Society. Vol. 662. American Mathematical Society. ISBN 978-0-8218-1003-3. <a href="978-0-8218-1003-3" target="_blank">978-0-8218-1003-3</a> <a href="#fnref:23" class="footnote-back-ref">↩</a></p></li> <li id="fn:24"><p>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. <a href="#fnref:24" class="footnote-back-ref">↩</a></p></li> <li id="fn:25"><p>Kharlampovich, Olga; Myasnikov, Alexei (1998). "Tarski's problem about the elementary theory of free groups has a positive solution". Electronic Research Announcements of the American Mathematical Society. 4 (14): 101–8. doi:10.1090/S1079-6762-98-00047-X. MR 1662319. <a href="https://doi.org/10.1090%2FS1079-6762-98-00047-X" target="_blank">https://doi.org/10.1090%2FS1079-6762-98-00047-X</a> <a href="#fnref:25" class="footnote-back-ref">↩</a></p></li> <li id="fn:26"><p>D. B. A. Epstein, J. W. Cannon, D. Holt, S. 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