Classification of subdivision rules for geometric groups of low dimension

Rushton, Brian

doi:10.1090/S1088-4173-2014-00269-0

Classification of subdivision rules for geometric groups of low dimension
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Conform. Geom. Dyn. 18 (2014), 171-191 Request permission

Abstract:

Subdivision rules create sequences of nested cell structures on CW-complexes, and they frequently arise from groups. In this paper, we develop several tools for classifying subdivision rules. We give a criterion for a subdivision rule to represent a Gromov hyperbolic space, and show that a subdivision rule for a hyperbolic group determines the Gromov boundary. We give a criterion for a subdivision rule to represent a Euclidean space of dimension less than 4. We also show that Nil and Sol geometries cannot be modeled by subdivision rules. We use these tools and previous theorems to classify the geometry of subdivision rules for low-dimensional geometric groups by the combinatorial properties of their subdivision rules.

References

Roger C. Alperin, Solvable groups of exponential growth and HNN extensions, Groups—Korea ’98 (Pusan), de Gruyter, Berlin, 2000, pp. 1–5. MR 1751083
J. W. Cannon, W. J. Floyd, and W. R. Parry, Finite subdivision rules, Conform. Geom. Dyn. 5 (2001), 153–196. MR 1875951, DOI 10.1090/S1088-4173-01-00055-8
James W. Cannon, The combinatorial Riemann mapping theorem, Acta Math. 173 (1994), no. 2, 155–234. MR 1301392, DOI 10.1007/BF02398434
J. W. Cannon, W. J. Floyd, and W. R. Parry, Sufficiently rich families of planar rings, Ann. Acad. Sci. Fenn. Math. 24 (1999), no. 2, 265–304. MR 1724092
J. W. Cannon, W. J. Floyd, and W. R. Parry, Expansion complexes for finite subdivision rules. I, Conform. Geom. Dyn. 10 (2006), 63–99. MR 2218641, DOI 10.1090/S1088-4173-06-00126-3
J. W. Cannon, W. J. Floyd, and W. R. Parry, Expansion complexes for finite subdivision rules. II, Conform. Geom. Dyn. 10 (2006), 326–354. MR 2268483, DOI 10.1090/S1088-4173-06-00127-5
J. W. Cannon and E. L. Swenson, Recognizing constant curvature discrete groups in dimension $3$, Trans. Amer. Math. Soc. 350 (1998), no. 2, 809–849. MR 1458317, DOI 10.1090/S0002-9947-98-02107-2
J. W. Cannon, W. J. Floyd, and W. R. Parry, Conformal modulus: the graph paper invariant or the conformal shape of an algorithm, Geometric group theory down under (Canberra, 1996) de Gruyter, Berlin, 1999, pp. 71–102. MR 1714840
Andrew Casson and Douglas Jungreis, Convergence groups and Seifert fibered $3$-manifolds, Invent. Math. 118 (1994), no. 3, 441–456. MR 1296353, DOI 10.1007/BF01231540
M. Coornaert, T. Delzant, and A. Papadopoulos, Géométrie et théorie des groupes, Lecture Notes in Mathematics, vol. 1441, Springer-Verlag, Berlin, 1990 (French). Les groupes hyperboliques de Gromov. [Gromov hyperbolic groups]; With an English summary. MR 1075994, DOI 10.1007/BFb0084913
Pierre de la Harpe, Topics in geometric group theory, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2000. MR 1786869
Cornelia Drutu and Michael Kapovich Preface, Lectures on geometric group theory, (2013), A book to be published in the AMS series “Colloquium Publications” in 2015.
D. Epstein, M.S. Paterson, J.W. Cannon, D.F. Holt, S.V. Levy, and W. P. Thurston, Word processing in groups, AK Peters, Ltd., 1992.
Benson Farb and Lee Mosher, Problems on the geometry of finitely generated solvable groups, Crystallographic groups and their generalizations (Kortrijk, 1999) Contemp. Math., vol. 262, Amer. Math. Soc., Providence, RI, 2000, pp. 121–134. MR 1796128, DOI 10.1090/conm/262/04170
F. T. Farrell and L. E. Jones, The lower algebraic $K$-theory of virtually infinite cyclic groups, $K$-Theory 9 (1995), no. 1, 13–30. MR 1340838, DOI 10.1007/BF00965457
Eric M. Freden, Negatively curved groups have the convergence property. I, Ann. Acad. Sci. Fenn. Ser. A I Math. 20 (1995), no. 2, 333–348. MR 1346817
David Gabai, Convergence groups are Fuchsian groups, Ann. of Math. (2) 136 (1992), no. 3, 447–510. MR 1189862, DOI 10.2307/2946597
Mikhael Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math. 53 (1981), 53–73. MR 623534, DOI 10.1007/BF02698687
Mikhael Gromov, Infinite groups as geometric objects, Proceedings of the International Congress of Mathematicians, vol. 1, 1984, p. 2.
M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263. MR 919829, DOI 10.1007/978-1-4613-9586-7_{3}
Ilya Kapovich and Nadia Benakli, Boundaries of hyperbolic groups, Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001) Contemp. Math., vol. 296, Amer. Math. Soc., Providence, RI, 2002, pp. 39–93. MR 1921706, DOI 10.1090/conm/296/05068
Avinoam Mann, How groups grow, London Mathematical Society Lecture Note Series, vol. 395, Cambridge University Press, Cambridge, 2012. MR 2894945
Grisha Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv preprint math/0211159 (2002).
—, Ricci flow with surgery on three-manifolds, arXiv preprint math/0303109 (2003).
Brian Rushton, Constructing subdivision rules from alternating links, Conform. Geom. Dyn. 14 (2010), 1–13. MR 2579862, DOI 10.1090/S1088-4173-09-00205-7
Brian Rushton, Constructing subdivision rules from polyhedra with identifications, Algebr. Geom. Topol. 12 (2012), no. 4, 1961–1992. MR 2994827, DOI 10.2140/agt.2012.12.1961
Brian Rushton, A finite subdivision rule for the n-dimensional torus, Geometriae Dedicata (2012), 1–12 (English).
Kenneth Stephenson, Introduction to circle packing, Cambridge University Press, Cambridge, 2005. The theory of discrete analytic functions. MR 2131318
Pekka Tukia, Homeomorphic conjugates of Fuchsian groups, J. Reine Angew. Math. 391 (1988), 1–54. MR 961162, DOI 10.1515/crll.1988.391.1
Jussi Väisälä, Gromov hyperbolic spaces, Expo. Math. 23 (2005), no. 3, 187–231. MR 2164775, DOI 10.1016/j.exmath.2005.01.010