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Convergence group
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<h2 id="formal-definition">Formal definition</h2> <p>Let Γ {\displaystyle \Gamma } be a group acting by homeomorphisms on a compact metrizable space M {\displaystyle M} . This action is called a <i>convergence action</i> or a <i>discrete convergence action</i> (and then Γ {\displaystyle \Gamma } is called a <i>convergence group</i> or a <i>discrete convergence group</i> for this action) if for every infinite distinct sequence of elements γ n ∈ Γ {\displaystyle \gamma _{n}\in \Gamma } there exist a subsequence γ n k , k = 1 , 2 , … {\displaystyle \gamma _{n_{k}},k=1,2,\dots } and points a , b ∈ M {\displaystyle a,b\in M} such that the maps γ n k | M ∖ { a } {\displaystyle \gamma _{n_{k}}{\big |}_{M\setminus \{a\}}} converge uniformly on compact subsets to the constant map sending M ∖ { a } {\displaystyle M\setminus \{a\}} to b {\displaystyle b} . Here converging uniformly on compact subsets means that for every open neighborhood U {\displaystyle U} of b {\displaystyle b} in M {\displaystyle M} and every compact K ⊂ M ∖ { a } {\displaystyle K\subset M\setminus \{a\}} there exists an index k 0 ≥ 1 {\displaystyle k_{0}\geq 1} such that for every k ≥ k 0 , {\displaystyle k\geq k_{0},} γ n k ( K ) ⊆ U {\displaystyle \gamma _{n_{k}}(K)\subseteq U} . Note that the "poles" a , b ∈ M {\displaystyle a,b\in M} associated with the subsequence γ n k {\displaystyle \gamma _{n_{k}}} are not required to be distinct. </p> <h3>Reformulation in terms of the action on distinct triples</h3> <p>The above definition of convergence group admits a useful equivalent reformulation in terms of the action of Γ {\displaystyle \Gamma } on the "space of distinct triples" of M {\displaystyle M} . For a set M {\displaystyle M} denote Θ ( M ) := M 3 ∖ Δ ( M ) {\displaystyle \Theta (M):=M^{3}\setminus \Delta (M)} , where Δ ( M ) = { ( a , b , c ) ∈ M 3 ∣ # { a , b , c } ≤ 2 } {\displaystyle \Delta (M)=\{(a,b,c)\in M^{3}\mid \#\{a,b,c\}\leq 2\}} . The set Θ ( M ) {\displaystyle \Theta (M)} is called the "space of distinct triples" for M {\displaystyle M} . </p><p>Then the following equivalence is known to hold:<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> </p><p>Let Γ {\displaystyle \Gamma } be a group acting by homeomorphisms on a compact metrizable space M {\displaystyle M} with at least two points. Then this action is a discrete convergence action <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> the induced action of Γ {\displaystyle \Gamma } on Θ ( M ) {\displaystyle \Theta (M)} is <a href="/facts/Properly_discontinuous_action/Vh9VtUUw">properly discontinuous</a>. </p> <h2 id="examples">Examples</h2> <ul><li>The action of a <a href="/facts/Kleinian_group/FBMCx9Ry">Kleinian group</a> Γ {\displaystyle \Gamma } on S 2 = ∂ H 3 {\displaystyle \mathbb {S} ^{2}=\partial \mathbb {H} ^{3}} by <a href="/facts/M%25C3%25B6bius_transformation/YEkTTLfK">Möbius transformations</a> is a convergence group action.</li> <li>The action of a <a href="/facts/Word-hyperbolic_group/yXGUebC4">word-hyperbolic group</a> G {\displaystyle G} by translations on its ideal boundary ∂ G {\displaystyle \partial G} is a convergence group action.</li> <li>The action of a <a href="/facts/Relatively_hyperbolic_group/u7336Qba">relatively hyperbolic group</a> G {\displaystyle G} by translations on its Bowditch boundary ∂ G {\displaystyle \partial G} is a convergence group action.</li> <li>Let X {\displaystyle X} be a proper geodesic <a href="/facts/%25CE%2594-hyperbolic_space/Daqo6Ah2">Gromov-hyperbolic</a> <a href="/facts/Metric_space/gnBBRXtc">metric space</a> and let Γ {\displaystyle \Gamma } be a group acting properly discontinuously by isometries on X {\displaystyle X} . Then the corresponding boundary action of Γ {\displaystyle \Gamma } on ∂ X {\displaystyle \partial X} is a discrete convergence action (Lemma 2.11 of <a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>).</li></ul> <h2 id="classification-of-elements-in-convergence-groups">Classification of elements in convergence groups</h2> <p>Let Γ {\displaystyle \Gamma } be a group acting by homeomorphisms on a compact metrizable space M {\displaystyle M} with at least three points, and let γ ∈ Γ {\displaystyle \gamma \in \Gamma } . Then it is known (Lemma 3.1 in <a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> or Lemma 6.2 in <a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a>) that exactly one of the following occurs: </p><p>(1) The element γ {\displaystyle \gamma } has finite order in Γ {\displaystyle \Gamma } ; in this case γ {\displaystyle \gamma } is called <i>elliptic</i>. </p><p>(2) The element γ {\displaystyle \gamma } has infinite order in Γ {\displaystyle \Gamma } and the fixed set Fix M ( γ ) {\displaystyle \operatorname {Fix} _{M}(\gamma )} is a single point; in this case γ {\displaystyle \gamma } is called <i>parabolic</i>. </p><p>(3) The element γ {\displaystyle \gamma } has infinite order in Γ {\displaystyle \Gamma } and the fixed set Fix M ( γ ) {\displaystyle \operatorname {Fix} _{M}(\gamma )} consists of two distinct points; in this case γ {\displaystyle \gamma } is called <i>loxodromic</i>. </p><p>Moreover, for every p ≠ 0 {\displaystyle p\neq 0} the elements γ {\displaystyle \gamma } and γ p {\displaystyle \gamma ^{p}} have the same type. Also in cases (2) and (3) Fix M ( γ ) = Fix M ( γ p ) {\displaystyle \operatorname {Fix} _{M}(\gamma )=\operatorname {Fix} _{M}(\gamma ^{p})} (where p ≠ 0 {\displaystyle p\neq 0} ) and the group ⟨ γ ⟩ {\displaystyle \langle \gamma \rangle } acts properly discontinuously on M ∖ Fix M ( γ ) {\displaystyle M\setminus \operatorname {Fix} _{M}(\gamma )} . Additionally, if γ {\displaystyle \gamma } is loxodromic, then ⟨ γ ⟩ {\displaystyle \langle \gamma \rangle } acts properly discontinuously and cocompactly on M ∖ Fix M ( γ ) {\displaystyle M\setminus \operatorname {Fix} _{M}(\gamma )} . </p><p>If γ ∈ Γ {\displaystyle \gamma \in \Gamma } is parabolic with a fixed point a ∈ M {\displaystyle a\in M} then for every x ∈ M {\displaystyle x\in M} one has lim n → ∞ γ n x = lim n → − ∞ γ n x = a {\displaystyle \lim _{n\to \infty }\gamma ^{n}x=\lim _{n\to -\infty }\gamma ^{n}x=a} If γ ∈ Γ {\displaystyle \gamma \in \Gamma } is loxodromic, then Fix M ( γ ) {\displaystyle \operatorname {Fix} _{M}(\gamma )} can be written as Fix M ( γ ) = { a − , a + } {\displaystyle \operatorname {Fix} _{M}(\gamma )=\{a_{-},a_{+}\}} so that for every x ∈ M ∖ { a − } {\displaystyle x\in M\setminus \{a_{-}\}} one has lim n → ∞ γ n x = a + {\displaystyle \lim _{n\to \infty }\gamma ^{n}x=a_{+}} and for every x ∈ M ∖ { a + } {\displaystyle x\in M\setminus \{a_{+}\}} one has lim n → − ∞ γ n x = a − {\displaystyle \lim _{n\to -\infty }\gamma ^{n}x=a_{-}} , and these convergences are uniform on compact subsets of M ∖ { a − , a + } {\displaystyle M\setminus \{a_{-},a_{+}\}} . </p> <h2 id="uniform-convergence-groups">Uniform convergence groups</h2> <p>A discrete convergence action of a group Γ {\displaystyle \Gamma } on a compact metrizable space M {\displaystyle M} is called <i>uniform</i> (in which case Γ {\displaystyle \Gamma } is called a <i>uniform convergence group</i>) if the action of Γ {\displaystyle \Gamma } on Θ ( M ) {\displaystyle \Theta (M)} is <a href="/facts/Cocompact_group_action/d9IoKywY">co-compact</a>. Thus Γ {\displaystyle \Gamma } is a <a href="/facts/Uniform_convergence/ecz9eNWn">uniform convergence</a> group if and only if its action on Θ ( M ) {\displaystyle \Theta (M)} is both properly discontinuous and co-compact. </p> <h3>Conical limit points</h3> <p>Let Γ {\displaystyle \Gamma } act on a compact metrizable space M {\displaystyle M} as a discrete convergence group. A point x ∈ M {\displaystyle x\in M} is called a <i>conical limit point</i> (sometimes also called a <i>radial limit point</i> or a <i>point of approximation</i>) if there exist an infinite sequence of distinct elements γ n ∈ Γ {\displaystyle \gamma _{n}\in \Gamma } and distinct points a , b ∈ M {\displaystyle a,b\in M} such that lim n → ∞ γ n x = a {\displaystyle \lim _{n\to \infty }\gamma _{n}x=a} and for every y ∈ M ∖ { x } {\displaystyle y\in M\setminus \{x\}} one has lim n → ∞ γ n y = b {\displaystyle \lim _{n\to \infty }\gamma _{n}y=b} . </p><p>An important result of <a href="/facts/Pekka_Tukia/D2EvQVIc">Tukia</a>,<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> also independently obtained by <a href="/facts/Brian_Bowditch/0BMSFmTp">Bowditch</a>,<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a><a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> states: </p><p>A discrete convergence group action of a group Γ {\displaystyle \Gamma } on a compact metrizable space M {\displaystyle M} is uniform if and only if every non-isolated point of M {\displaystyle M} is a conical limit point. </p> <h3>Word-hyperbolic groups and their boundaries</h3> <p>It was already observed by Gromov<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> that the natural action by translations of a <a href="/facts/Word-hyperbolic_group/yXGUebC4">word-hyperbolic group</a> G {\displaystyle G} on its boundary ∂ G {\displaystyle \partial G} is a uniform convergence action (see<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> for a formal proof). Bowditch<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> proved an important converse, thus obtaining a topological characterization of word-hyperbolic groups: </p><p>Theorem. Let G {\displaystyle G} act as a discrete uniform convergence group on a compact metrizable space M {\displaystyle M} with no isolated points. Then the group G {\displaystyle G} is word-hyperbolic and there exists a G {\displaystyle G} -equivariant homeomorphism M → ∂ G {\displaystyle M\to \partial G} . </p> <h2 id="convergence-actions-on-the-circle">Convergence actions on the circle</h2> <p>An isometric action of a group G {\displaystyle G} on the <a href="/facts/Hyperbolic_plane/DjTMKdgy">hyperbolic plane</a> H 2 {\displaystyle \mathbb {H} ^{2}} is called <a href="/facts/Geometric_group_action/1qH8ngRu"><i>geometric</i></a> if this action is properly discontinuous and cocompact. Every geometric action of G {\displaystyle G} on H 2 {\displaystyle \mathbb {H} ^{2}} induces a uniform convergence action of G {\displaystyle G} on S 1 = ∂ H 2 ≈ ∂ G {\displaystyle \mathbb {S} ^{1}=\partial H^{2}\approx \partial G} . An important result of Tukia (1986),<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> <a href="/facts/David_Gabai/HPR734x6">Gabai</a> (1992),<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a> Casson–Jungreis (1994),<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> and Freden (1995)<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a> shows that the converse also holds: </p><p>Theorem. If G {\displaystyle G} is a group acting as a discrete uniform convergence group on S 1 {\displaystyle \mathbb {S} ^{1}} then this action is topologically conjugate to an action induced by a geometric action of G {\displaystyle G} on H 2 {\displaystyle \mathbb {H} ^{2}} by isometries. </p><p>Note that whenever G {\displaystyle G} acts geometrically on H 2 {\displaystyle \mathbb {H} ^{2}} , the group G {\displaystyle G} is <a href="/facts/Virtually/mVQqmv2B">virtually</a> a hyperbolic surface group, that is, G {\displaystyle G} contains a finite index subgroup isomorphic to the fundamental group of a closed hyperbolic surface. </p> <h2 id="convergence-actions-on-the-2-sphere">Convergence actions on the 2-sphere</h2> <p>One of the equivalent reformulations of <a href="/facts/Cannon%2527s_conjecture/M40dDJUm">Cannon's conjecture</a>, (posed by <a href="/facts/James_W._Cannon/9U7rATUF">James W. Cannon</a>,<a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a> although an earlier and more general conjecture, reducing to the Cannon conjecture for compact type, was given by Gaven J. Martin and Richard K. Skora <a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a>) </p><p>These conjectures are in terms of word-hyperbolic groups with boundaries homeomorphic to S 2 {\displaystyle \mathbb {S} ^{2}} , says that if G {\displaystyle G} is a group acting as a discrete uniform convergence group on S 2 {\displaystyle \mathbb {S} ^{2}} then this action is topologically conjugate to an action induced by a <a href="/facts/Geometric_group_action/1qH8ngRu">geometric action</a> of G {\displaystyle G} on H 3 {\displaystyle \mathbb {H} ^{3}} by isometries. These conjectures still remains open. </p> <h2 id="applications-and-further-generalizations">Applications and further generalizations</h2> <ul><li>Yaman gave a characterization of <a href="/facts/Relatively_hyperbolic_group/u7336Qba">relatively hyperbolic groups</a> in terms of convergence actions,<a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a> generalizing Bowditch's characterization of word-hyperbolic groups as uniform convergence groups.</li> <li>One can consider more general versions of group actions with "convergence property" without the discreteness assumption.<a class="footnote-ref" id="fnref:19" href="#fn:19"><sup>19</sup></a></li> <li>The most general version of the notion of <a href="/facts/Cannon%25E2%2580%2593Thurston_map/JhNQLJHS">Cannon–Thurston map</a>, originally defined in the context of Kleinian and word-hyperbolic groups, can be defined and studied in the context of setting of convergence groups.<a class="footnote-ref" id="fnref:20" href="#fn:20"><sup>20</sup></a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Gehring, F. W.; Martin, G. J. (1987). "Discrete quasiconformal groups I". Proceedings of the London Mathematical Society. 55 (2): 331–358. doi:10.1093/plms/s3-55_2.331. hdl:2027.42/135296. <a href="/wiki/Proceedings_of_the_London_Mathematical_Society" target="_blank">/wiki/Proceedings_of_the_London_Mathematical_Society</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Bowditch, B. H. (1999). "Convergence groups and configuration spaces". Geometric group theory down under (Canberra, 1996). De Gruyter Proceedings in Mathematics. de Gruyter, Berlin. pp. 23–54. doi:10.1515/9783110806861.23. ISBN 9783110806861. <a href="9783110806861" target="_blank">9783110806861</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Bowditch, B. H. (1999). "Convergence groups and configuration spaces". Geometric group theory down under (Canberra, 1996). De Gruyter Proceedings in Mathematics. de Gruyter, Berlin. pp. 23–54. doi:10.1515/9783110806861.23. ISBN 9783110806861. <a href="9783110806861" target="_blank">9783110806861</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Bowditch, B. H. (1999). "Convergence groups and configuration spaces". Geometric group theory down under (Canberra, 1996). De Gruyter Proceedings in Mathematics. de Gruyter, Berlin. pp. 23–54. doi:10.1515/9783110806861.23. ISBN 9783110806861. <a href="9783110806861" target="_blank">9783110806861</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Bowditch, B. H. (1999). "Treelike structures arising from continua and convergence groups". Memoirs of the American Mathematical Society. 139 (662). doi:10.1090/memo/0662. <a href="/wiki/Brian_Bowditch" target="_blank">/wiki/Brian_Bowditch</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Tukia, Pekka (1998). "Conical limit points and uniform convergence groups". Journal für die reine und angewandte Mathematik. 1998 (501): 71–98. doi:10.1515/crll.1998.081. <a href="/wiki/Journal_f%C3%BCr_die_reine_und_angewandte_Mathematik" target="_blank">/wiki/Journal_f%C3%BCr_die_reine_und_angewandte_Mathematik</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Bowditch, B. H. (1999). "Convergence groups and configuration spaces". Geometric group theory down under (Canberra, 1996). De Gruyter Proceedings in Mathematics. de Gruyter, Berlin. pp. 23–54. doi:10.1515/9783110806861.23. ISBN 9783110806861. <a href="9783110806861" target="_blank">9783110806861</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Bowditch, Brian H. (1998). "A topological characterisation of hyperbolic groups". Journal of the American Mathematical Society. 11 (3): 643–667. doi:10.1090/S0894-0347-98-00264-1. <a href="https://doi.org/10.1090%2FS0894-0347-98-00264-1" target="_blank">https://doi.org/10.1090%2FS0894-0347-98-00264-1</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Gromov, Mikhail (1987). "Hyperbolic groups". In Gersten, Steve M. (ed.). Essays in group theory. Mathematical Sciences Research Institute Publications. Vol. 8. New York: Springer. pp. 75–263. doi:10.1007/978-1-4613-9586-7_3. ISBN 0-387-96618-8. MR 0919829. <a href="0-387-96618-8" target="_blank">0-387-96618-8</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Bowditch, B. H. (1999). "Convergence groups and configuration spaces". Geometric group theory down under (Canberra, 1996). De Gruyter Proceedings in Mathematics. de Gruyter, Berlin. pp. 23–54. doi:10.1515/9783110806861.23. ISBN 9783110806861. <a href="9783110806861" target="_blank">9783110806861</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Bowditch, Brian H. (1998). "A topological characterisation of hyperbolic groups". Journal of the American Mathematical Society. 11 (3): 643–667. doi:10.1090/S0894-0347-98-00264-1. <a href="https://doi.org/10.1090%2FS0894-0347-98-00264-1" target="_blank">https://doi.org/10.1090%2FS0894-0347-98-00264-1</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Tukia, Pekka (1986). "On quasiconformal groups". Journal d'Analyse Mathématique. 46: 318–346. doi:10.1007/BF02796595. <a href="/wiki/Pekka_Tukia" target="_blank">/wiki/Pekka_Tukia</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>Gabai, Davis (1992). "Convergence groups are Fuchsian groups". Annals of Mathematics. Second series. 136 (3): 447–510. doi:10.2307/2946597. JSTOR 2946597. <a href="/wiki/David_Gabai" target="_blank">/wiki/David_Gabai</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>Casson, Andrew; Jungreis, Douglas (1994). "Convergence groups and Seifert fibered 3-manifolds". Inventiones Mathematicae. 118 (3): 441–456. Bibcode:1994InMat.118..441C. doi:10.1007/BF01231540. <a href="/wiki/Inventiones_Mathematicae" target="_blank">/wiki/Inventiones_Mathematicae</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>Freden, Eric M. (1995). "Negatively curved groups have the convergence property I" (PDF). Annales Academiae Scientiarum Fennicae. Series A. 20 (2): 333–348. Retrieved September 12, 2022. <a href="https://www.acadsci.fi/mathematica/Vol20/freden.pdf" target="_blank">https://www.acadsci.fi/mathematica/Vol20/freden.pdf</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> <li id="fn:16"><p>Cannon, James W. (1991). "The theory of negatively curved spaces and groups" (PDF). Ergodic theory, symbolic dynamics, and hyperbolic spaces (Trieste, 1989). Oxford Sci. Publ., Oxford Univ. Press, New York. pp. 315–369. Retrieved September 12, 2022. <a href="http://mmontee.people.sites.carleton.edu/Cannon_Negatively_Curved.pdf" target="_blank">http://mmontee.people.sites.carleton.edu/Cannon_Negatively_Curved.pdf</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></p></li> <li id="fn:17"><p>{{Citation last1=Martin | first1=Gaven J. | last2=Skora | first2=Richard K. | title=Group Actions of the 2-Sphere | journal=American Journal of Mathematics | volume=111 | pages=387–402 | publisher=The Johns Hopkins University Press | date=1989 }} <a href="/wiki/American_Journal_of_Mathematics" target="_blank">/wiki/American_Journal_of_Mathematics</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></p></li> <li id="fn:18"><p>Yaman, Asli (2004). "A topological characterisation of relatively hyperbolic groups". Journal für die reine und angewandte Mathematik. 2004 (566): 41–89. doi:10.1515/crll.2004.007. <a href="/wiki/Journal_f%C3%BCr_die_reine_und_angewandte_Mathematik" target="_blank">/wiki/Journal_f%C3%BCr_die_reine_und_angewandte_Mathematik</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></p></li> <li id="fn:19"><p>Gerasimov, Victor (2009). "Expansive convergence groups are relatively hyperbolic". Geometric and Functional Analysis. 19 (1): 137–169. doi:10.1007/s00039-009-0718-7. <a href="/wiki/Geometric_and_Functional_Analysis" target="_blank">/wiki/Geometric_and_Functional_Analysis</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></p></li> <li id="fn:20"><p>Jeon, Woojin; Kapovich, Ilya; Leininger, Christopher; Ohshika, Ken'ichi (2016). "Conical limit points and the Cannon-Thurston map". Conformal Geometry and Dynamics. 20 (4): 58–80. arXiv:1401.2638. doi:10.1090/ecgd/294. <a href="/wiki/Ilya_Kapovich" target="_blank">/wiki/Ilya_Kapovich</a> <a href="#fnref:20" class="footnote-back-ref">↩</a></p></li> </ol>