We generalize a class of groups introduced by Herbert Abels to produce examples of virtually torsion free groups that have Bredon-finiteness length m − 1 and classical finiteness length n − 1 for all 0 < m ≤ n.
The proof illustrates how Bredon-finiteness properties can be verified using geometric methods and a version of Brown’s criterion due to Martin Fluch and the author.
Keywords: finiteness properties, Bredon homology, Abels's groups, horospheres, arithmetic groups, buildings
Witzel, Stefan 1
@article{10_2140_agt_2013_13_3447,
author = {Witzel, Stefan},
title = {Abels{\textquoteright}s groups revisited},
journal = {Algebraic and Geometric Topology},
pages = {3447--3467},
year = {2013},
volume = {13},
number = {6},
doi = {10.2140/agt.2013.13.3447},
url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2013.13.3447/}
}
Witzel, Stefan. Abels’s groups revisited. Algebraic and Geometric Topology, Tome 13 (2013) no. 6, pp. 3447-3467. doi: 10.2140/agt.2013.13.3447
[1] , An example of a finitely presented solvable group, from: "Homological group theory" (editor C T C Wall), London Math. Soc. Lecture Note Ser. 36, Cambridge Univ. Press (1979) 205
[2] , , Finiteness properties of solvable $S$–arithmetic groups: An example, from: "Proceedings of the Northwestern conference on cohomology of groups", J. Pure Appl. Algebra 44 (1987) 77
[3] , , Buildings: Theory and applications, Graduate Texts in Mathematics 248, Springer (2008)
[4] , , Morse theory and finiteness properties of groups, Invent. Math. 129 (1997) 445
[5] , Equivariant cohomology theories, Lecture Notes in Mathematics 34, Springer (1967)
[6] , Finiteness properties for subgroups of $\mathrm{GL}(n,\mathbf Z)$, Math. Ann. 317 (2000) 629
[7] , Presentations for groups acting on simply-connected complexes, J. Pure Appl. Algebra 32 (1984) 1
[8] , Finiteness properties of groups, from: "Proceedings of the Northwestern conference on cohomology of groups", J. Pure Appl. Algebra 44 (1987) 45
[9] , , Schémas en groupes et immeubles des groupes classiques sur un corps local, Bull. Soc. Math. France 112 (1984) 259
[10] , , Connectivity properties of horospheres in Euclidean buildings and applications to finiteness properties of discrete groups, Invent. Math. 185 (2011) 395
[11] , , Brown's criterion in Bredon homology, to appear in Homology, Homotopy Appl.
[12] , Finite generation of $K$-groups of a curve over a finite field (after Daniel Quillen), from: "Algebraic $K$-theory, Part I" (editor R K Dennis), Lecture Notes in Math. 966, Springer (1982) 69
[13] , , On finite extensions of arithmetic groups, C. R. Acad. Sci. Paris Sér. I Math. 325 (1997) 1153
[14] , , , Centralisers of finite subgroups in soluble groups of type $\mathrm{FP}_n$, Forum Math. 23 (2011) 99
[15] , , , Cohomological finiteness conditions for elementary amenable groups, J. Reine Angew. Math. 637 (2009) 49
[16] , , Some groups of type $VF$, Invent. Math. 151 (2003) 135
[17] , Transformation groups and algebraic $K$–theory, Lecture Notes in Mathematics 1408, Springer (1989)
[18] , , On the universal space for group actions with compact isotropy, from: "Geometry and topology: Aarhus" (editors K Grove, I H Madsen, E K Pedersen), Contemp. Math. 258, Amer. Math. Soc. (2000) 293
[19] , The theory of algebraic linear groups and periodic groups, Izv. Akad. Nauk SSSR Ser. Mat. 30 (1966) 573
[20] , , On finite group actions on reductive groups and buildings, Invent. Math. 147 (2002) 545
[21] , Lectures on buildings, Perspectives in Mathematics 7, Academic Press (1989)
[22] , Spherical subcomplexes of spherical buildings, Geom. Topol. 17 (2013) 531
[23] , Finitely presented soluble groups, from: "Group theory" (editors K W Gruenberg, J E Roseblade), Academic Press (1984) 257
[24] , Reductive groups over local fields, from: "Automorphic forms, representations and $L$–functions, Part 1" (editors A Borel, W Casselman), Proc. Sympos. Pure Math. 33, Amer. Math. Soc. (1979) 29
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