Coarse Alexander duality for pairs and applications

Hruska, G Christopher¹ ; Stark, Emily² ; Trần, Hùng Công³

¹Department of Mathematical Sciences, University of Wisconsin–Milwaukee, Milwaukee, WI, United States
²Department of Mathematics, Wesleyan University, Middletown, CT, United States
³FCCI Insurance Group, Sarasota, FL, United States

Algebraic and Geometric Topology, Tome 25 (2025) no. 4, pp. 1999-2035

Résumé

For a group G (of type F) acting properly on a coarse Poincaré duality space X, Kapovich and Kleiner introduced a coarse version of Alexander duality between G and its complement in X. More precisely, the cohomology of G with group ring coefficients is dual to a certain Čech homology group of the family of increasing neighborhoods of a G-orbit in X. This duality applies more generally to coarse embeddings of certain contractible simplicial complexes into coarse PD ⁡ (n) spaces. In this paper we introduce a relative version of this Čech homology that satisfies the Eilenberg–Steenrod exactness axiom, and we prove a relative version of coarse Alexander duality.

As an application we provide a detailed proof of the following result, first stated by Kapovich and Kleiner. Given a 2-complex formed by gluing k halfplanes along their boundary lines and a coarse embedding into a contractible 3-manifold, the complement consists of k deep components that are arranged cyclically in a pattern called a Jordan cycle. We use the Jordan cycle as an invariant in proving the existence of a 3-manifold group that is virtually Kleinian but not itself Kleinian.

Détail

DOI : 10.2140/agt.2025.25.1999

Keywords: Alexander duality, coarse $\mathrm{PD}(n)$ space, Kleinian group

Bibliographie

[1] M Bestvina, Local homology properties of boundaries of groups, Michigan Math. J. 43 (1996) 123 | DOI

[2] M Bestvina, M Kapovich, B Kleiner, Van Kampen’s embedding obstruction for discrete groups, Invent. Math. 150 (2002) 219 | DOI

[3] J Block, S Weinberger, Large scale homology theories and geometry, from: "Geometric topology" (editor W H Kazez), AMS/IP Stud. Adv. Math. 2.1, Amer. Math. Soc. (1997) 522 | DOI

[4] K S Brown, Cohomology of groups, 87, Springer (1982) | DOI

[5] J Dydak, Local n-connectivity of quotient spaces and one-point compactifications, from: "Shape theory and geometric topology" (editors S Mardešić, J Segal), Lecture Notes in Math. 870, Springer (1981) 48 | DOI

[6] S Eilenberg, N Steenrod, Foundations of algebraic topology, Princeton Univ. Press (1952)

[7] B Farb, R Schwartz, The large-scale geometry of Hilbert modular groups, J. Differential Geom. 44 (1996) 435

[8] R Geoghegan, The shape of a group: connections between shape theory and the homology of groups, from: "Geometric and algebraic topology" (editors H Toruńczyk, S Jackowski, S Spie.z), Banach Center Publ. 18, PWN (1986) 271

[9] R Geoghegan, Topological methods in group theory, 243, Springer (2008) | DOI

[10] R Geoghegan, E Swenson, On semistability of CAT(0) groups, Groups Geom. Dyn. 13 (2019) 695 | DOI

[11] C R Guilbault, Ends, shapes, and boundaries in manifold topology and geometric group theory, from: "Topology and geometric group theory" (editors M W Davis, J Fowler, J F Lafont, I J Leary), Springer Proc. Math. Stat. 184, Springer (2016) 45 | DOI

[12] A Hatcher, Algebraic topology, Cambridge Univ. Press (2002)

[13] G C Hruska, B Kleiner, Hadamard spaces with isolated flats, Geom. Topol. 9 (2005) 1501 | DOI

[14] G C Hruska, E Stark, H C Tran, Surface group amalgams that (don’t) act on 3-manifolds, Amer. J. Math. 142 (2020) 885 | DOI

[15] M Kapovich, Hyperbolic manifolds and discrete groups, 183, Birkhäuser (2001) | DOI

[16] M Kapovich, B Kleiner, Hyperbolic groups with low-dimensional boundary, Ann. Sci. École Norm. Sup. 33 (2000) 647 | DOI

[17] M Kapovich, B Kleiner, Coarse Alexander duality and duality groups, J. Differential Geom. 69 (2005) 279

[18] M Kapovich, B Leeb, On asymptotic cones and quasi-isometry classes of fundamental groups of 3-manifolds, Geom. Funct. Anal. 5 (1995) 582 | DOI

[19] M Kapovich, B Leeb, Quasi-isometries preserve the geometric decomposition of Haken manifolds, Invent. Math. 128 (1997) 393 | DOI

[20] S Mardešić, J Segal, Shape theory: the inverse system approach, 26, North-Holland (1982)

[21] J Milnor, On the Steenrod homology theory, from: "Novikov conjectures, index theorems and rigidity, I" (editors S C Ferry, A Ranicki, J Rosenberg), Lond. Math. Soc. Lect. Note Ser. 226, Cambridge Univ. Press (1995) 79 | DOI

[22] W D Neumann, G A Swarup, Canonical decompositions of 3-manifolds, Geom. Topol. 1 (1997) 21 | DOI

[23] J Roe, Lectures on coarse geometry, 31, Amer. Math. Soc. (2003) | DOI

[24] E H Spanier, Algebraic topology, McGraw-Hill (1966) | DOI