Distinguishing geometries using finite quotients

Wilton, Henry¹ ; Zalesskii, Pavel²

¹ DPMMS, Cambridge University, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 0WB, United Kingdom
² Department of Mathematics, University of Brasília, 70910-9000 Brasília, Brazil

Geometry & topology, Tome 21 (2017) no. 1, pp. 345-384.

Résumé

We prove that the profinite completion of the fundamental group of a compact $3$–manifold $M$ satisfies a Tits alternative: if a closed subgroup $H$ does not contain a free pro-$p$ subgroup for any $p$, then $H$ is virtually soluble, and furthermore of a very particular form. In particular, the profinite completion of the fundamental group of a closed, hyperbolic $3$–manifold does not contain a subgroup isomorphic to ${\widehat{\mathbb{Z}}}^2$. This gives a profinite characterization of hyperbolicity among irreducible $3$–manifolds. We also characterize Seifert fibred $3$–manifolds as precisely those for which the profinite completion of the fundamental group has a nontrivial procyclic normal subgroup. Our techniques also apply to hyperbolic, virtually special groups, in the sense of Haglund and Wise. Finally, we prove that every finitely generated pro-$p$ subgroup of the profinite completion of a torsion-free, hyperbolic, virtually special group is free pro-$p$.

Détail

DOI : 10.2140/gt.2017.21.345

Classification : 57N10, 20E26, 57M05
Keywords: $3$–manifolds, profinite completions

Bibliographie

[1] I Agol, The virtual Haken conjecture, Doc. Math. 18 (2013) 1045

[2] I Agol, D Groves, J F Manning, Residual finiteness, QCERF and fillings of hyperbolic groups, Geom. Topol. 13 (2009) 1043 | DOI

[3] M Aschenbrenner, S Friedl, H Wilton, 3–manifold groups, European Mathematical Society (2015) | DOI

[4] M Boggi, P Zalesskii, A restricted Magnus property for profinite surface groups, preprint (2014)

[5] M R Bridson, M D E Conder, A W Reid, Determining Fuchsian groups by their finite quotients, Israel Journal of Mathematics 214 (2016) 1 | DOI

[6] D Calegari, M H Freedman, K Walker, Positivity of the universal pairing in 3 dimensions, J. Amer. Math. Soc. 23 (2010) 107 | DOI

[7] A Casson, D Jungreis, Convergence groups and Seifert fibered 3–manifolds, Invent. Math. 118 (1994) 441 | DOI

[8] W Dicks, Groups, trees and projective modules, 790, Springer (1980)

[9] L Funar, Torus bundles not distinguished by TQFT invariants, Geom. Topol. 17 (2013) 2289 | DOI

[10] D Gabai, Convergence groups are Fuchsian groups, Ann. of Math. 136 (1992) 447 | DOI

[11] R Gitik, M Mitra, E Rips, M Sageev, Widths of subgroups, Trans. Amer. Math. Soc. 350 (1998) 321 | DOI

[12] D Groves, J F Manning, Quasiconvexity and Dehn filling, in preparation

[13] F Grunewald, A Jaikin-Zapirain, P A Zalesskii, Cohomological goodness and the profinite completion of Bianchi groups, Duke Math. J. 144 (2008) 53 | DOI

[14] F Haglund, D T Wise, Special cube complexes, Geom. Funct. Anal. 17 (2008) 1551 | DOI

[15] F Haglund, D T Wise, Coxeter groups are virtually special, Adv. Math. 224 (2010) 1890 | DOI

[16] F Haglund, D T Wise, A combination theorem for special cube complexes, Ann. of Math. 176 (2012) 1427 | DOI

[17] E Hamilton, Abelian subgroup separability of Haken 3–manifolds and closed hyperbolic n–orbifolds, Proc. London Math. Soc. 83 (2001) 626 | DOI

[18] E Hamilton, Finite quotients of rings and applications to subgroup separability of linear groups, Trans. Amer. Math. Soc. 357 (2005) 1995 | DOI

[19] E Hamilton, H Wilton, P A Zalesskii, Separability of double cosets and conjugacy classes in 3–manifold groups, J. Lond. Math. Soc. 87 (2013) 269 | DOI

[20] J Hempel, Residual finiteness for 3–manifolds, from: "Combinatorial group theory and topology" (editors S M Gersten, J R Stallings), Ann. of Math. Stud. 111, Princeton Univ. Press (1987) 379

[21] J Hempel, Some 3–manifold groups with the same finite quotients, preprint (2014)

[22] W Herfort, P A Zalesskii, Virtually free pro-p groups whose torsion elements have finite centralizer, Bull. Lond. Math. Soc. 40 (2008) 929 | DOI

[23] W Herfort, P Zalesskii, Virtually free pro-p groups, Publ. Math. Inst. Hautes Études Sci. 118 (2013) 193 | DOI

[24] W Herfort, P Zalesskii, T Zapata, Splitting theorems for pro-p groups acting on pro-p trees, Selecta Mathematica 22 (2016) 1245 | DOI

[25] G C Hruska, Relative hyperbolicity and relative quasiconvexity for countable groups, Algebr. Geom. Topol. 10 (2010) 1807 | DOI

[26] T Hsu, D T Wise, Cubulating malnormal amalgams, Invent. Math. 199 (2015) 293 | DOI

[27] J Kahn, V Markovic, Immersing almost geodesic surfaces in a closed hyperbolic three manifold, Ann. of Math. 175 (2012) 1127 | DOI

[28] B Kleiner, J Lott, Notes on Perelman’s papers, Geom. Topol. 12 (2008) 2587 | DOI

[29] D D Long, A W Reid, Grothendieck’s problem for 3–manifold groups, Groups Geom. Dyn. 5 (2011) 479 | DOI

[30] G Mess, The Seifert conjecture and groups which are coarse quasiisometric to planes, unpublished (1987)

[31] A Minasyan, Hereditary conjugacy separability of right-angled Artin groups and its applications, Groups Geom. Dyn. 6 (2012) 335 | DOI

[32] A Minasyan, P Zalesskii, One-relator groups with torsion are conjugacy separable, J. Algebra 382 (2013) 39 | DOI

[33] J Morgan, G Tian, Ricci flow and the Poincaré conjecture, 3, Amer. Math. Soc. (2007)

[34] J Morgan, G Tian, The geometrization conjecture, 5, Amer. Math. Soc. (2014)

[35] B B Newman, Some results on one-relator groups, Bull. Amer. Math. Soc. 74 (1968) 568 | DOI

[36] Y Ollivier, D T Wise, Cubulating random groups at density less than , Trans. Amer. Math. Soc. 363 (2011) 4701 | DOI

[37] G Perelman, The entropy formula for the Ricci flow and its geometric applications, preprint (2002)

[38] G Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, preprint (2003)

[39] G Perelman, Ricci flow with surgery on three-manifolds, preprint (2003)

[40] A W Reid, Profinite properties of discrete groups, from: "Groups St Andrews 2013" (editors C M Campbell, M R Quick, E F Robertson), London Math. Soc. Lecture Note Ser. 422, Cambridge Univ. Press (2015) 73 | DOI

[41] L Ribes, P Zalesskii, Pro-p trees and applications, from: "New horizons in pro-p groups" (editors M du Sautoy, D Segal, A Shalev), Progr. Math. 184, Birkhäuser (2000) 75 | DOI

[42] L Ribes, P Zalesskii, Profinite groups, 40, Springer (2010) | DOI

[43] P Scott, Subgroups of surface groups are almost geometric, J. London Math. Soc. 17 (1978) 555 | DOI

[44] P Scott, The geometries of 3–manifolds, Bull. London Math. Soc. 15 (1983) 401 | DOI

[45] J P Serre, Arbres, amalgames, SL2, 46, Société Mathématique de France (1977)

[46] P F Stebe, Conjugacy separability of groups of integer matrices, Proc. Amer. Math. Soc. 32 (1972) 1 | DOI

[47] P Tukia, Homeomorphic conjugates of Fuchsian groups, J. Reine Angew. Math. 391 (1988) 1 | DOI

[48] H Wilton, P Zalesskii, Profinite properties of graph manifolds, Geom. Dedicata 147 (2010) 29 | DOI

[49] D T Wise, Cubulating small cancellation groups, Geom. Funct. Anal. 14 (2004) 150 | DOI

[50] D T Wise, The structure of groups with a quasi-convex hierarchy, preprint (2012)

[51] P A Zalesskiĭ, Geometric characterization of free constructions of profinite groups, Sibirsk. Mat. Zh. 30 (1989) 73 | DOI

[52] P A Zalesskiĭ, Profinite groups, without free nonabelian pro-p–subgroups, that act on trees, Mat. Sb. 181 (1990) 57

[53] P A Zalesskiĭ, Open subgroups of free profinite products, from: "Proceedings of the International Conference on Algebra" (editors L A Bokut’, Y L Ershov, A I Kostrikin), Contemp. Math. 131, Amer. Math. Soc. (1992) 473

[54] P A Zalesskiĭ, O V Mel’Nikov, Subgroups of profinite groups acting on trees, Mat. Sb. 135(177) (1988) 419

[55] P A Zalesskiĭ, O V Mel’Nikov, Fundamental groups of graphs of profinite groups, Algebra i Analiz 1 (1989) 117