A Higman embedding preserving asphericity

Sapir, Mark

doi:10.1090/S0894-0347-2013-00776-6

A Higman embedding preserving asphericity
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by Mark Sapir

J. Amer. Math. Soc. 27 (2014), 1-42

DOI: https://doi.org/10.1090/S0894-0347-2013-00776-6

Published electronically: July 9, 2013

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Abstract:

We prove that every finitely generated group with recursive aspherical presentation embeds into a group with finite aspherical presentation. This and several known facts about groups and manifolds imply that there exists a 4-dimensional closed aspherical manifold $M$ such that the fundamental group $\pi _1(M)$ coarsely contains an expander. Thus it has infinite asymptotic dimension, is not coarsely embeddable into a Hilbert space, does not satisfy G. Yu’s property A, and does not satisfy the Baum-Connes conjecture with coefficients. Closed aspherical manifolds with any of these properties were previously unknown.

References

G. N. Arzhantseva, T. Delzant, Examples of random groups, preprint, 2008.
J.-C. Birget, A. Yu. Ol′shanskii, E. Rips, and M. V. Sapir, Isoperimetric functions of groups and computational complexity of the word problem, Ann. of Math. (2) 156 (2002), no. 2, 467–518. MR 1933724, DOI 10.2307/3597196
G. Baumslag, E. Dyer, and C. F. Miller III, On the integral homology of finitely presented groups, Topology 22 (1983), no. 1, 27–46. MR 682058, DOI 10.1016/0040-9383(83)90044-7
Oleg Bogopolski and Enric Ventura, A recursive presentation for Mihailova’s subgroup, Groups Geom. Dyn. 4 (2010), no. 3, 407–417. MR 2653968, DOI 10.4171/GGD/88
Ian M. Chiswell, Donald J. Collins, and Johannes Huebschmann, Aspherical group presentations, Math. Z. 178 (1981), no. 1, 1–36. MR 627092, DOI 10.1007/BF01218369
D. J. Collins and J. Huebschmann, Spherical diagrams and identities among relations, Math. Ann. 261 (1982), no. 2, 155–183. MR 675732, DOI 10.1007/BF01456216
Rémi Coulon, Small Cancellation Theory and Burnside problem, preprint, arXiv:1302.6933, to appear in IJAC.
Michael W. Davis, Exotic aspherical manifolds, Topology of high-dimensional manifolds, No. 1, 2 (Trieste, 2001) ICTP Lect. Notes, vol. 9, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2002, pp. 371–404. MR 1937019
Michael W. Davis, The geometry and topology of Coxeter groups, London Mathematical Society Monographs Series, vol. 32, Princeton University Press, Princeton, NJ, 2008. MR 2360474
Alexander Dranishnikov, Open problems in asymptotic dimension theory, preprint, http://www.aimath.org/pggt/Asymptotic Dimension, 2008.
M. Gromov, Asymptotic invariants of infinite groups, Geometric group theory, Vol. 2 (Sussex, 1991) London Math. Soc. Lecture Note Ser., vol. 182, Cambridge Univ. Press, Cambridge, 1993, pp. 1–295. MR 1253544
M. Gromov, Random walk in random groups, Geom. Funct. Anal. 13 (2003), no. 1, 73–146. MR 1978492, DOI 10.1007/s000390300002
Erik Guentner, Romain Tessera, and Guoliang Yu, A notion of geometric complexity and its application to topological rigidity, Invent. Math. 189 (2012), no. 2, 315–357. MR 2947546, DOI 10.1007/s00222-011-0366-z
P. J. Heawood, Map-Colour Theorems, Quarterly Journal of Mathematics, Oxford 24, 1890, 332–338.
G. Higman, Subgroups of finitely presented groups, Proc. Roy. Soc. London Ser. A 262 (1961), 455–475. MR 130286, DOI 10.1098/rspa.1961.0132
N. Higson, V. Lafforgue, and G. Skandalis, Counterexamples to the Baum-Connes conjecture, Geom. Funct. Anal. 12 (2002), no. 2, 330–354. MR 1911663, DOI 10.1007/s00039-002-8249-5
Wolfgang Lück, Survey on aspherical manifolds, European Congress of Mathematics, Eur. Math. Soc., Zürich, 2010, pp. 53–82. MR 2648321, DOI 10.4171/077-1/4
Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89, Springer-Verlag, Berlin-New York, 1977. MR 0577064
Alexey Muranov, Finitely generated infinite simple groups of infinite square width and vanishing stable commutator length, J. Topol. Anal. 2 (2010), no. 3, 341–384. MR 2718128, DOI 10.1142/S1793525310000380
A. Yu. Ol′shanskiĭ, Geometry of defining relations in groups, Mathematics and its Applications (Soviet Series), vol. 70, Kluwer Academic Publishers Group, Dordrecht, 1991. Translated from the 1989 Russian original by Yu. A. Bakhturin. MR 1191619, DOI 10.1007/978-94-011-3618-1
A. Yu. Ol′shanskiĭ, $\textrm {SQ}$-universality of hyperbolic groups, Mat. Sb. 186 (1995), no. 8, 119–132 (Russian, with Russian summary); English transl., Sb. Math. 186 (1995), no. 8, 1199–1211. MR 1357360, DOI 10.1070/SM1995v186n08ABEH000063
Alexander Yu. Ol′shanskii, Denis V. Osin, and Mark V. Sapir, Lacunary hyperbolic groups, Geom. Topol. 13 (2009), no. 4, 2051–2140. With an appendix by Michael Kapovich and Bruce Kleiner. MR 2507115, DOI 10.2140/gt.2009.13.2051
A. Yu. Ol′shanskii and M. V. Sapir, The conjugacy problem and Higman embeddings, Mem. Amer. Math. Soc. 170 (2004), no. 804, viii+133. MR 2052958, DOI 10.1090/memo/0804
A. Yu. Ol′shanskii and M. V. Sapir, Groups with small Dehn functions and bipartite chord diagrams, Geom. Funct. Anal. 16 (2006), no. 6, 1324–1376. MR 2276542, DOI 10.1007/s00039-006-0580-9
Joseph J. Rotman, An introduction to the theory of groups, 4th ed., Graduate Texts in Mathematics, vol. 148, Springer-Verlag, New York, 1995. MR 1307623, DOI 10.1007/978-1-4612-4176-8
Mark Sapir, Algorithmic and asymptotic properties of groups, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 223–244. MR 2275595
Mark Sapir, Asymptotic invariants, complexity of groups and related problems, Bull. Math. Sci. 1 (2011), no. 2, 277–364. MR 2901003, DOI 10.1007/s13373-011-0008-1
Mark Sapir, Non-commutative combinatorial algebra, Springer, to appear, 2014.
Mark V. Sapir, Jean-Camille Birget, and Eliyahu Rips, Isoperimetric and isodiametric functions of groups, Ann. of Math. (2) 156 (2002), no. 2, 345–466. MR 1933723, DOI 10.2307/3597195
Rufus Willett, Property A and graphs with large girth, J. Topol. Anal. 3 (2011), no. 3, 377–384. MR 2831267, DOI 10.1142/S179352531100057X
Guoliang Yu, The Novikov conjecture for groups with finite asymptotic dimension, Ann. of Math. (2) 147 (1998), no. 2, 325–355. MR 1626745, DOI 10.2307/121011
Guoliang Yu, The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space, Invent. Math. 139 (2000), no. 1, 201–240. MR 1728880, DOI 10.1007/s002229900032