The Farrell–Jones conjecture for normally poly-free groups

Brück, Benjamin; Kielak, Dawid; Wu, Xiaolei

doi:10.1090/proc/15357

The Farrell–Jones conjecture for normally poly-free groups
HTML articles powered by AMS MathViewer

by Benjamin Brück, Dawid Kielak and Xiaolei Wu

Proc. Amer. Math. Soc. 149 (2021), 2349-2356

DOI: https://doi.org/10.1090/proc/15357

Published electronically: March 23, 2021

PDF | Request permission

Abstract:

We prove the $K$- and $L$-theoretic Farrell–Jones Conjecture with coefficients in an additive category for every normally poly-free group, in particular for even Artin groups of FC-type, and for all groups of the form $A\rtimes \mathbb {Z}$ where $A$ is a right-angled Artin group. Our proof relies on the work of Bestvina–Fujiwara–Wigglesworth on the Farrell–Jones Conjecture for free-by-cyclic groups.

References

C. S. Aravinda, F. T. Farrell, and S. K. Roushon, Algebraic $K$-theory of pure braid groups, Asian J. Math. 4 (2000), no. 2, 337–343. MR 1797585, DOI 10.4310/AJM.2000.v4.n2.a4
A. Bartels, Coarse flow spaces for relatively hyperbolic groups, Compos. Math. 153 (2017), no. 4, 745–779. MR 3631229, DOI 10.1112/S0010437X16008216
Arthur Bartels and Wolfgang Lück, The Borel conjecture for hyperbolic and $\textrm {CAT}(0)$-groups, Ann. of Math. (2) 175 (2012), no. 2, 631–689. MR 2993750, DOI 10.4007/annals.2012.175.2.5
Mladen Bestvina, Non-positively curved aspects of Artin groups of finite type, Geom. Topol. 3 (1999), 269–302. MR 1714913, DOI 10.2140/gt.1999.3.269
M. Bestvina, K. Fujiwara, and D. Wigglesworth, Farrell–Jones conjecture for free-by-cyclic groups, Preprint, 2019, arXiv:1906.00069.
Rubén Blasco-García, Conchita Martínez-Pérez, and Luis Paris, Poly-freeness of even Artin groups of FC type, Groups Geom. Dyn. 13 (2019), no. 1, 309–325. MR 3900773, DOI 10.4171/GGD/486
N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1337, Hermann, Paris, 1968 (French). MR 0240238
Ruth Charney and Karen Vogtmann, Finiteness properties of automorphism groups of right-angled Artin groups, Bull. Lond. Math. Soc. 41 (2009), no. 1, 94–102. MR 2481994, DOI 10.1112/blms/bdn108
Nils-Edvin Enkelmann, Wolfgang Lück, Malte Pieper, Mark Ullmann, and Christoph Winges, On the Farrell-Jones conjecture for Waldhausen’s $A$-theory, Geom. Topol. 22 (2018), no. 6, 3321–3394. MR 3858766, DOI 10.2140/gt.2018.22.3321
F. Thomas Farrell and Xiaolei Wu, The isomorphism conjecture for solvable groups in Waldhausen’s A-theory, J. Topol. Anal. 11 (2019), no. 2, 405–426. MR 3958927, DOI 10.1142/S1793525319500195
Herbert Federer and Bjarni Jónsson, Some properties of free groups, Trans. Amer. Math. Soc. 68 (1950), 1–27. MR 32643, DOI 10.1090/S0002-9947-1950-0032643-4
Giovanni Gandini, Sebastian Meinert, and Henrik Rüping, The Farrell-Jones conjecture for fundamental groups of graphs of abelian groups, Groups Geom. Dyn. 9 (2015), no. 3, 783–792. MR 3420543, DOI 10.4171/GGD/327
J. Huang and D. Osajda, Helly meets Garside and Artin, Preprint, 2019, arXiv:1904.09436.
Daniel Kasprowski, Mark Ullmann, Christian Wegner, and Christoph Winges, The $A$-theoretic Farrell-Jones conjecture for virtually solvable groups, Bull. Lond. Math. Soc. 50 (2018), no. 2, 219–228. MR 3830115, DOI 10.1112/blms.12131
Svenja Knopf, Acylindrical actions on trees and the Farrell-Jones conjecture, Groups Geom. Dyn. 13 (2019), no. 2, 633–676. MR 3950645, DOI 10.4171/GGD/499
Michael R. Laurence, A generating set for the automorphism group of a graph group, J. London Math. Soc. (2) 52 (1995), no. 2, 318–334. MR 1356145, DOI 10.1112/jlms/52.2.318
W. Lück, Assembly maps, Preprint, 2018, arXiv:1805.00226.
Jon McCammond, The mysterious geometry of Artin groups, Winter Braids Lect. Notes 4 (2017), no. Winter Braids VII (Caen, 2017), Exp. No. 1, 30. MR 3922033, DOI 10.5802/wbln.17
Holger Reich and Marco Varisco, Algebraic $K$-theory, assembly maps, controlled algebra, and trace methods, Space—time—matter, De Gruyter, Berlin, 2018, pp. 1–50. MR 3792301
S. K. Roushon, A certain structure of Artin groups and the isomorphism conjecture, Preprint, 2018, arXiv:1811.11589.
Jean-Pierre Serre, Trees, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. Translated from the French original by John Stillwell; Corrected 2nd printing of the 1980 English translation. MR 1954121
Herman Servatius, Automorphisms of graph groups, J. Algebra 126 (1989), no. 1, 34–60. MR 1023285, DOI 10.1016/0021-8693(89)90319-0
H. Van der Lek, The homotopy type of complex hyperplane complements, Ph.D. thesis. Nijmegen, 1983.
Christian Wegner, The $K$-theoretic Farrell-Jones conjecture for CAT(0)-groups, Proc. Amer. Math. Soc. 140 (2012), no. 3, 779–793. MR 2869063, DOI 10.1090/S0002-9939-2011-11150-X
Christian Wegner, The Farrell-Jones conjecture for virtually solvable groups, J. Topol. 8 (2015), no. 4, 975–1016. MR 3431666, DOI 10.1112/jtopol/jtv026
Xiaolei Wu, Poly-freeness of Artin groups and the Farrell-Jones Conjecture, Preprint, arXiv:1912.04350.