no. 2
Action rigidity for free products of hyperbolic manifold groups
[Rigidité des actions des produits libres de groupes de variétés hyperboliques]
Stark, Emily R.1 ; Woodhouse, Daniel J.2
1 Wesleyan University Department of Mathematics 265 Church Street Middletown, CT 06459 (USA)
2 University of Oxford Mathematical Institute Andrew Wiles Building, Woodstock Road Oxford, OX2 6GG (United Kingdon)
Annales de l'Institut Fourier, Tome 74 (2024) no. 2, pp. 503-544.

Deux groupes ont la même géométrie de modèle s’ils agissent proprement et cocompactement par isométries sur un même espace métrique propre et géodésique. Nous étudions les produits libres de réseaux uniformes dans les groupes d’isométries d’espaces symétriques de rang 1 et prouvons que, dans chaque classe de quasi-isométries, les groupes résiduellement finis qui ont la même géométrie de modèle sont abstraitement commensurables. Notre résultat donne les premiers exemples de groupes hyperboliques qui sont quasi-isométriques mais qui n’ont virtuellement pas la même géométrie de modèle. Un élément important de la preuve est une généralisation d’un théorème de Leighton sur les revêtements finis communs des graphes. Le théorème principal utilise la notion de finitude résiduelle, et nous montrons que les extensions finies de réseaux uniformes des espaces symétriques de rang 1 qui ne sont pas résiduellement finis produisent des contre-exemples.

Two groups have a common model geometry if they act properly and cocompactly by isometries on the same proper geodesic metric space. We consider free products of uniform lattices in isometry groups of rank-1 symmetric spaces and prove, within each quasi-isometry class, that residually finite groups that have a common model geometry are abstractly commensurable. Our result gives the first examples of hyperbolic groups that are quasi-isometric but do not virtually have a common model geometry. An important component of the proof is a generalization of Leighton’s graph covering theorem. The main theorem depends on residual finiteness, and we show that finite extensions of uniform lattices in rank-1 symmetric spaces that are not residually finite would give counterexamples.

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DOI : 10.5802/aif.3585
Classification : 20F65, 20F67, 20E06, 57M07, 57M10
Keywords: Gromov Hyperbolicity, Residual finiteness, Bass–Serre theory, Action rigidity.
Mot clés : Hyperbolicité de Gromov, groupes résiduellement finis, théorie de Bass–Serre, rigidité des actions.

Stark, Emily R. 1 ; Woodhouse, Daniel J. 2

1 Wesleyan University Department of Mathematics 265 Church Street Middletown, CT 06459 (USA)
2 University of Oxford Mathematical Institute Andrew Wiles Building, Woodstock Road Oxford, OX2 6GG (United Kingdon)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Stark, Emily R.; Woodhouse, Daniel J. Action rigidity for free products of hyperbolic manifold groups. Annales de l'Institut Fourier, Tome 74 (2024) no. 2, pp. 503-544. doi : 10.5802/aif.3585. https://aif.centre-mersenne.org/articles/10.5802/aif.3585/

[1] Agol, Ian The virtual Haken conjecture, Doc. Math., Volume 18 (2013), pp. 1045-1087 (with an appendix by Agol, Daniel Groves, and Jason Manning) | DOI | MR | Zbl

[2] Bass, Hyman Covering theory for graphs of groups, J. Pure Appl. Algebra, Volume 89 (1993) no. 1-2, pp. 3-47 https://ezproxy-prd.bodleian.ox.ac.uk:2102/10.1016/0022-4049(93)90085-8 | DOI | MR | Zbl

[3] Bass, Hyman; Kulkarni, Ravi Uniform tree lattices, J. Amer. Math. Soc., Volume 3 (1990) no. 4, pp. 843-902 | DOI | MR | Zbl

[4] Baumslag, Benjamin; Tretkoff, Marvin Residually finite HNN extensions, Comm. Algebra, Volume 6 (1978) no. 2, pp. 179-194 | DOI | MR | Zbl

[5] Baumslag, Gilbert On the residual finiteness of generalised free products of nilpotent groups, Trans. Amer. Math. Soc., Volume 106 (1963), pp. 193-209 | DOI | MR | Zbl

[6] Behrstock, Jason A.; Januszkiewicz, Tadeusz; Neumann, Walter D. Commensurability and QI classification of free products of finitely generated abelian groups, Proc. Amer. Math. Soc., Volume 137 (2009) no. 3, pp. 811-813 | DOI | MR | Zbl

[7] Behrstock, Jason A.; Neumann, Walter D. Quasi-isometric classification of non-geometric 3-manifold groups, J. Reine Angew. Math., Volume 669 (2012), pp. 101-120 | DOI | MR | Zbl

[8] Bergeron, Nicolas; Wise, Daniel T. A boundary criterion for cubulation, Amer. J. Math., Volume 134 (2012) no. 3, pp. 843-859 https://ezproxy-prd.bodleian.ox.ac.uk:2102/10.1353/ajm.2012.0020 | DOI | MR | Zbl

[9] Bestvina, Mladen; Feighn, Mark Bounding the complexity of simplicial group actions on trees, Invent. Math., Volume 103 (1991) no. 3, pp. 449-469 | DOI | MR | Zbl

[10] Bowditch, Brian H. A course on geometric group theory, MSJ Memoirs, 16, Mathematical Society of Japan, Tokyo, 2006, x+104 pages | DOI | MR | Zbl

[11] Bridson, Martin R.; Haefliger, André Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften, 319, Springer-Verlag, Berlin, 1999, xxii+643 pages | DOI | MR | Zbl

[12] Burger, Marc; Mozes, Shahar Lattices in product of trees, Inst. Hautes Études Sci. Publ. Math. (2000) no. 92, p. 151-194 (2001) | DOI | Numdam | MR | Zbl

[13] Casson, Andrew; Jungreis, Douglas Convergence groups and Seifert fibered 3-manifolds, Invent. Math., Volume 118 (1994) no. 3, pp. 441-456 https://ezproxy-prd.bodleian.ox.ac.uk:2094/10.1007/BF01231540 | DOI | MR | Zbl

[14] Chow, Richard Groups quasi-isometric to complex hyperbolic space, Trans. Amer. Math. Soc., Volume 348 (1996) no. 5, pp. 1757-1769 https://ezproxy-prd.bodleian.ox.ac.uk:4563/10.1090/S0002-9947-96-01522-X | DOI | MR | Zbl

[15] Cornulier, Yves; de la Harpe, Pierre Metric geometry of locally compact groups, EMS Tracts in Mathematics, 25, European Mathematical Society (EMS), Zürich, 2016, viii+235 pages (winner of the 2016 EMS Monograph Award) | DOI | MR | Zbl

[16] Das, Kajal; Tessera, Romain Integrable measure equivalence and the central extension of surface groups, Groups Geom. Dyn., Volume 10 (2016) no. 3, pp. 965-983 https://ezproxy-prd.bodleian.ox.ac.uk:2102/10.4171/GGD/373 | DOI | MR | Zbl

[17] Deligne, Pierre Extensions centrales non résiduellement finies de groupes arithmétiques, C. R. Acad. Sci. Paris Sér. A-B, Volume 287 (1978) no. 4, p. A203-A208 | MR | Zbl

[18] Druţu, Cornelia; Kapovich, Michael Geometric group theory, American Mathematical Society Colloquium Publications, 63, American Mathematical Society, Providence, RI, 2018, xx+819 pages (with an appendix by Bogdan Nica) | DOI | MR | Zbl

[19] Dunwoody, M. J. The accessibility of finitely presented groups, Invent. Math., Volume 81 (1985) no. 3, pp. 449-457 | DOI | MR | Zbl

[20] Friedl, Stefan; Park, JungHwan; Petri, Bram; Raimbault, Jean; Ray, Arunima Nonmanifold hyperbolic groups of high cohomological dimension (https://arxiv.org/abs/1807.09861)

[21] Gabai, David Convergence groups are Fuchsian groups, Ann. of Math. (2), Volume 136 (1992) no. 3, pp. 447-510 | DOI | MR | Zbl

[22] Gaboriau, Damien Invariants l 2 de relations d’équivalence et de groupes, Publ. Math. Inst. Hautes études Sci. (2002) no. 95, pp. 93-150 | DOI | MR | Zbl

[23] Gaboriau, Damien; Noûs, Camille On the top-dimensional ℓ 2 -Betti numbers, Ann. Fac. Sci. Toulouse Math. (6), Volume 30 (2021) no. 5, pp. 1121-1137 https://doi-org.ezproxy.wesleyan.edu/10.5802/afst.1695 | DOI | MR | Zbl

[24] Gromov, M. Hyperbolic groups, Essays in group theory (Math. Sci. Res. Inst. Publ.), Volume 8, Springer, New York, 1987, pp. 75-263 | DOI | MR | Zbl

[25] Gromov, M.; Piatetski-Shapiro, I. Nonarithmetic groups in Lobachevsky spaces, Inst. Hautes études Sci. Publ. Math. (1988) no. 66, pp. 93-103 | DOI | MR | Zbl

[26] Grunewald, F.; Jaikin-Zapirain, A.; Zalesskii, P. A. Cohomological goodness and the profinite completion of Bianchi groups, Duke Math. J., Volume 144 (2008) no. 1, pp. 53-72 https://ezproxy-prd.bodleian.ox.ac.uk:2102/10.1215/00127094-2008-031 | DOI | MR | Zbl

[27] Grunewald, Fritz; Jaikin-Zapirain, Andrei; Pinto, Aline G. S.; Zalesskii, Pavel A. Normal subgroups of profinite groups of non-negative deficiency, J. Pure Appl. Algebra, Volume 218 (2014) no. 5, pp. 804-828 https://ezproxy-prd.bodleian.ox.ac.uk:2102/10.1016/j.jpaa.2013.10.003 | DOI | MR | Zbl

[28] Hill, Richard M. Non-residually finite extensions of arithmetic groups, Res. Number Theory, Volume 5 (2019) no. 1, p. Paper No. 2, 27 https://ezproxy-prd.bodleian.ox.ac.uk:2095/10.1007/s40993-018-0140-z | DOI | MR | Zbl

[29] Hinkkanen, A. Uniformly quasisymmetric groups, Proc. London Math. Soc. (3), Volume 51 (1985) no. 2, pp. 318-338 | DOI | MR | Zbl

[30] Hinkkanen, A. The structure of certain quasisymmetric groups, Mem. Amer. Math. Soc., Volume 83 (1990) no. 422, p. iv+87 | DOI | MR | Zbl

[31] Hruska, G. Christopher; Wise, Daniel T. Packing subgroups in relatively hyperbolic groups, Geom. Topol., Volume 13 (2009) no. 4, pp. 1945-1988 | DOI | MR | Zbl

[32] Kleiner, Bruce; Leeb, Bernhard Groups quasi-isometric to symmetric spaces, Comm. Anal. Geom., Volume 9 (2001) no. 2, pp. 239-260 https://ezproxy-prd.bodleian.ox.ac.uk:2094/10.4310/CAG.2001.v9.n2.a1 | DOI | MR | Zbl

[33] Kleiner, Bruce; Leeb, Bernhard Induced quasi-actions: a remark, Proc. Amer. Math. Soc., Volume 137 (2009) no. 5, pp. 1561-1567 https://ezproxy-prd.bodleian.ox.ac.uk:2094/10.1090/S0002-9939-08-09742-6 | DOI | MR | Zbl

[34] Leighton, Frank Thomson Finite common coverings of graphs, J. Combin. Theory Ser. B, Volume 33 (1982) no. 3, pp. 231-238 | DOI | MR | Zbl

[35] Maclachlan, Colin; Reid, Alan W. The arithmetic of hyperbolic 3-manifolds, Graduate Texts in Mathematics, 219, Springer-Verlag, New York, 2003, xiv+463 pages | DOI | MR | Zbl

[36] Markovic, Vladimir Quasisymmetric groups, J. Amer. Math. Soc., Volume 19 (2006) no. 3, pp. 673-715 | DOI | MR | Zbl

[37] Martin, Alexandre; Światkowski, Jacek Infinitely-ended hyperbolic groups with homeomorphic Gromov boundaries, J. Group Theory, Volume 18 (2015) no. 2, pp. 273-289 | DOI | MR | Zbl

[38] Mosher, Lee; Sageev, Michah; Whyte, Kevin Quasi-actions on trees. I. Bounded valence, Ann. of Math. (2), Volume 158 (2003) no. 1, pp. 115-164 | DOI | MR | Zbl

[39] Mostow, G. D. Strong rigidity of locally symmetric spaces, Annals of Mathematics Studies, 78, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1973, v+195 pages | DOI | MR | Zbl

[40] Neumann, Walter D.; Reid, Alan W. Arithmetic of hyperbolic manifolds, Topology ’90 (Columbus, OH, 1990) (Ohio State Univ. Math. Res. Inst. Publ.), Volume 1, de Gruyter, Berlin, 1992, pp. 273-310 | MR | Zbl

[41] Pansu, Pierre Métriques de Carnot–Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. of Math. (2), Volume 129 (1989) no. 1, pp. 1-60 https://ezproxy-prd.bodleian.ox.ac.uk:4563/10.2307/1971484 | DOI | MR | Zbl

[42] Papasoglu, Panos; Whyte, Kevin Quasi-isometries between groups with infinitely many ends, Comment. Math. Helv., Volume 77 (2002) no. 1, pp. 133-144 | DOI | MR | Zbl

[43] Scott, Peter; Wall, Terry Topological methods in group theory, Homological group theory (Proc. Sympos., Durham, 1977) (London Math. Soc. Lecture Note Ser.), Volume 36, Cambridge Univ. Press, Cambridge-New York, 1979, pp. 137-203 | DOI | MR | Zbl

[44] Serre, Jean-Pierre Trees, Springer-Verlag, Berlin-New York, 1980, ix+142 pages (translated from the French by John Stillwell) | DOI | MR | Zbl

[45] Shepherd, Sam; Gardam, Giles; Woodhouse, Daniel J. Two generalisations of Leighton’s Theorem (https://arxiv.org/abs/1908.00830)

[46] Stallings, John R. On torsion-free groups with infinitely many ends, Ann. of Math. (2), Volume 88 (1968), pp. 312-334 | DOI | MR | Zbl

[47] Stark, Emily; Woodhouse, Daniel J. Quasi-isometric groups with no common model geometry, J. Lond. Math. Soc. (2), Volume 99 (2019) no. 3, pp. 853-871 https://doi-org.ezproxy.wesleyan.edu/10.1112/jlms.12189 | DOI | MR | Zbl

[48] Swenson, Eric L. Quasi-convex groups of isometries of negatively curved spaces, Topology Appl., Volume 110 (2001) no. 1, pp. 119-129 Geometric topology and geometric group theory (Milwaukee, WI, 1997) | DOI | MR | Zbl

[49] Tretkoff, Marvin The residual finiteness of certain amalgamated free products, Math. Z., Volume 132 (1973), pp. 179-182 | DOI | MR | Zbl

[50] Tukia, Pekka On quasiconformal groups, J. Analyse Math., Volume 46 (1986), pp. 318-346 | DOI | MR | Zbl

[51] Whyte, Kevin Amenability, bi-Lipschitz equivalence, and the von Neumann conjecture, Duke Math. J., Volume 99 (1999) no. 1, pp. 93-112 | DOI | MR | Zbl

[52] Wise, Dani The structure of groups with a quasiconvex hierarchy (http://comet.lehman.cuny.edu/behrstock/cbms/program.html, unpublished manuscript)

[53] Woodhouse, Daniel J. Revisiting Leighton’s theorem with the Haar measure, Math. Proc. Cambridge Philos. Soc., Volume 170 (2021) no. 3, pp. 615-623 https://doi-org.ezproxy.wesleyan.edu/10.1017/S0305004119000550 | DOI | MR | Zbl

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