On planar Cayley graphs and Kleinian groups

Georgakopoulos, Agelos

doi:10.1090/tran/8026

On planar Cayley graphs and Kleinian groups
HTML articles powered by AMS MathViewer

by Agelos Georgakopoulos PDF

Trans. Amer. Math. Soc. 373 (2020), 4649-4684

Abstract:

Let $G$ be a finitely generated group acting faithfully and properly discontinuously by homeomorphisms on a planar surface $X \subseteq \mathbb {S}^2$. We prove that $G$ admits such an action that is in addition co-compact, provided we can replace $X$ by another surface $Y \subseteq \mathbb {S}^2$.

We also prove that if a group $H$ has a finitely generated Cayley (multi-) graph $C$ equivariantly embeddable in $\mathbb {S}^2$, then $C$ can be chosen so as to have no infinite path on the boundary of a face.

The proofs of these facts are intertwined, and the classes of groups they define coincide. In the orientation-preserving case they are exactly the (isomorphism types of) finitely generated Kleinian function groups. We construct a finitely generated planar Cayley graph whose group is not in this class.

In passing, we observe that the Freudenthal compactification of every planar surface is homeomorphic to the sphere.

References

G. N. Arzhantseva and P.-A. Cherix, On the Cayley graph of a generic finitely presented group, Bull. Belg. Math. Soc. Simon Stevin 11 (2004), no. 4, 589–601. MR 2115727, DOI 10.36045/bbms/1102689123
László Babai, Some applications of graph contractions, J. Graph Theory 1 (1977), no. 2, 125–130. Special issue dedicated to Paul Turán. MR 460171, DOI 10.1002/jgt.3190010207
László Babai, The growth rate of vertex-transitive planar graphs, Proceedings of the Eighth Annual ACM-SIAM Symposium on Discrete Algorithms (New Orleans, LA, 1997) ACM, New York, 1997, pp. 564–573. MR 1447704
Joseph Ernest Borzellino, Riemannian geometry of orbifolds, ProQuest LLC, Ann Arbor, MI, 1992. Thesis (Ph.D.)–University of California, Los Angeles. MR 2687544
B. H. Bowditch and G. Mess, A $4$-dimensional Kleinian group, Trans. Amer. Math. Soc. 344 (1994), no. 1, 391–405. MR 1240944, DOI 10.1090/S0002-9947-1994-1240944-6
L. E. J. Brouwer, On the structure of perfect sets of points, Proc. Koninklijke Akademie van Wetenschappen 12 (1910), 785–794.
Morton Brown, A proof of the generalized Schoenflies theorem, Bull. Amer. Math. Soc. 66 (1960), 74–76. MR 117695, DOI 10.1090/S0002-9904-1960-10400-4
Warren Dicks and M. J. Dunwoody, Groups acting on graphs, Cambridge Studies in Advanced Mathematics, vol. 17, Cambridge University Press, Cambridge, 1989. MR 1001965
Reinhard Diestel, Graph theory, 3rd ed., Springer-Verlag, 2005. Electronic edition available at: http://www.math.uni-hamburg.de/home/diestel/books/graph.theory.
Carl Droms, Infinite-ended groups with planar Cayley graphs, J. Group Theory 9 (2006), no. 4, 487–496. MR 2243241, DOI 10.1515/JGT.2006.032
Carl Droms, Brigitte Servatius, and Herman Servatius, Connectivity and planarity of Cayley graphs, Beiträge Algebra Geom. 39 (1998), no. 2, 269–282. MR 1642723
M. J. Dunwoody, The accessibility of finitely presented groups, Invent. Math. 81 (1985), no. 3, 449–457. MR 807066, DOI 10.1007/BF01388581
M. J. Dunwoody, Planar graphs and covers, Preprint available at http://www.personal. soton.ac.uk/mjd7/, 2009.
Éric Charpentier, Étienne Ghys, and Annick Lesne (eds.), The scientific legacy of Poincaré, History of Mathematics, vol. 36, American Mathematical Society, Providence, RI, 2010. Translated from the 2006 French original by Joshua Bowman. MR 2605614, DOI 10.1090/hmath/036
Felix Klein, Vergleichende Betrachtungen über neuere geometrische Forschungen, Verlag von Andreas Deichert, Erlangen, 1872.
Agelos Georgakopoulos, Graph topologies induced by edge lengths, Discrete Math. 311 (2011), no. 15, 1523–1542. MR 2800976, DOI 10.1016/j.disc.2011.02.012
Agelos Georgakopoulos, The planar cubic Cayley graphs, Mem. Amer. Math. Soc. 250 (2017), no. 1190, v+82. MR 3709725, DOI 10.1090/memo/1190
A. Georgakopoulos and M. Hamann, The planar Cayley graphs are effectively enumerable I: consistently planar graphs, To appear in Combinatorica.
A. Georgakopoulos and M. Hamann, The planar Cayley graphs are effectively enumerable II, Preprint 2018.
Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. MR 1867354
Heinz Hopf, Enden offener Räume und unendliche diskontinuierliche Gruppen, Comment. Math. Helv. 16 (1944), 81–100 (German). MR 10267, DOI 10.1007/BF02568567
Wilfried Imrich, On Whitney’s theorem on the unique embeddability of $3$-connected planar graphs, Recent advances in graph theory (Proc. Second Czechoslovak Sympos., Prague, 1974) Academia, Prague, 1975, pp. 303–306. (loose errata). MR 0384588
M. Kapovich, A note on properly discontinuous actions, https://www.math.ucdavis.edu /~kapovich/EPR/prop-disc.pdf.
B. Krön, Infinite faces and ends of almost transitive plane graphs, Preprint.
K. Kuratowski, Topology. Vol. II, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1968. New edition, revised and augmented; Translated from the French by A. Kirkor. MR 0259835
Henry Levinson and Bernard Maskit, Special embeddings of Cayley diagrams, J. Combinatorial Theory Ser. B 18 (1975), 12–17. MR 384598, DOI 10.1093/comjnl/18.3.287
Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1977 edition. MR 1812024, DOI 10.1007/978-3-642-61896-3
A. Marden, Geometrically finite Kleinian groups and their deformation spaces, Discrete groups and automorphic functions (Proc. Conf., Cambridge, 1975) Academic Press, London, 1977, pp. 259–293. MR 0494117
A. Marden, Outer circles, Cambridge University Press, Cambridge, 2007. An introduction to hyperbolic 3-manifolds. MR 2355387, DOI 10.1017/CBO9780511618918
H. Maschke, The Representation of Finite Groups, Especially of the Rotation Groups of the Regular Bodies of Three-and Four-Dimensional Space, by Cayley’s Color Diagrams, Amer. J. Math. 18 (1896), no. 2, 156–194. MR 1505708, DOI 10.2307/2369680
Bernard Maskit, On the classification of Kleinian groups. I. Koebe groups, Acta Math. 135 (1975), no. 3-4, 249–270. MR 444942, DOI 10.1007/BF02392021
Bernard Maskit, Kleinian groups, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 287, Springer-Verlag, Berlin, 1988. MR 959135
Barry Mazur, On embeddings of spheres, Bull. Amer. Math. Soc. 65 (1959), 59–65. MR 117693, DOI 10.1090/S0002-9904-1959-10274-3
Bojan Mohar, Tree amalgamation of graphs and tessellations of the Cantor sphere, J. Combin. Theory Ser. B 96 (2006), no. 5, 740–753. MR 2236509, DOI 10.1016/j.jctb.2006.02.003
Edwin E. Moise, Geometric topology in dimensions $2$ and $3$, Graduate Texts in Mathematics, Vol. 47, Springer-Verlag, New York-Heidelberg, 1977. MR 0488059, DOI 10.1007/978-1-4612-9906-6
Ken’ichi Ohshika, Discrete groups, Translations of Mathematical Monographs, vol. 207, American Mathematical Society, Providence, RI, 2002. Translated from the 1998 Japanese original by the author; Iwanami Series in Modern Mathematics. MR 1862839, DOI 10.1090/mmono/207
T. Radö, Über den Begriff der Riemannschen Fläche, Acta Sci. Math. (Szeged), 2 (1925), no. 1, 96–114.
Ian Richards, On the classification of noncompact surfaces, Trans. Amer. Math. Soc. 106 (1963), 259–269. MR 143186, DOI 10.1090/S0002-9947-1963-0143186-0
R. Bruce Richter and Carsten Thomassen, 3-connected planar spaces uniquely embed in the sphere, Trans. Amer. Math. Soc. 354 (2002), no. 11, 4585–4595. MR 1926890, DOI 10.1090/S0002-9947-02-03052-0
Peter Scott, The geometries of $3$-manifolds, Bull. London Math. Soc. 15 (1983), no. 5, 401–487. MR 705527, DOI 10.1112/blms/15.5.401
Caroline Series, A crash course on Kleinian groups, Rend. Istit. Mat. Univ. Trieste 37 (2005), no. 1-2, 1–38 (2006). MR 2227047
Carsten Thomassen, Tilings of the torus and the Klein bottle and vertex-transitive graphs on a fixed surface, Trans. Amer. Math. Soc. 323 (1991), no. 2, 605–635. MR 1040045, DOI 10.1090/S0002-9947-1991-1040045-3
Carsten Thomassen, The Jordan-Schönflies theorem and the classification of surfaces, Amer. Math. Monthly 99 (1992), no. 2, 116–130. MR 1144352, DOI 10.2307/2324180
Carsten Thomassen and Antoine Vella, Graph-like continua, augmenting arcs, and Menger’s theorem, Combinatorica 28 (2008), no. 5, 595–623. MR 2501250, DOI 10.1007/s00493-008-2342-9
William P. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 3, 357–381. MR 648524, DOI 10.1090/S0273-0979-1982-15003-0
William Thurston, The geometry and topology of three-manifolds, Princeton lecture notes, 1980.
Thomas W. Tucker, Finite groups acting on surfaces and the genus of a group, J. Combin. Theory Ser. B 34 (1983), no. 1, 82–98. MR 701174, DOI 10.1016/0095-8956(83)90009-6
Hassler Whitney, Congruent Graphs and the Connectivity of Graphs, Amer. J. Math. 54 (1932), no. 1, 150–168. MR 1506881, DOI 10.2307/2371086
Heiner Zieschang, Elmar Vogt, and Hans-Dieter Coldewey, Surfaces and planar discontinuous groups, Lecture Notes in Mathematics, vol. 835, Springer, Berlin, 1980. Translated from the German by John Stillwell. MR 606743, DOI 10.1007/BFb0089692