## Algebraic & Geometric Topology

### Abels's groups revisited

Stefan Witzel

#### Abstract

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.

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.

#### Article information

Source
Algebr. Geom. Topol., Volume 13, Number 6 (2013), 3447-3467.

Dates
Accepted: 27 February 2013
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715738

Digital Object Identifier
doi:10.2140/agt.2013.13.3447

Mathematical Reviews number (MathSciNet)
MR3248739

Zentralblatt MATH identifier
1297.20054

#### Citation

Witzel, Stefan. Abels's groups revisited. Algebr. Geom. Topol. 13 (2013), no. 6, 3447--3467. doi:10.2140/agt.2013.13.3447. https://projecteuclid.org/euclid.agt/1513715738

#### References

• H Abels, An example of a finitely presented solvable group, from: “Homological group theory”, (C T C Wall, editor), London Math. Soc. Lecture Note Ser. 36, Cambridge Univ. Press (1979) 205–211
• H Abels, K S Brown, 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) 77–83
• P Abramenko, K S Brown, Buildings: Theory and applications, Graduate Texts in Mathematics 248, Springer, New York (2008)
• M Bestvina, N Brady, Morse theory and finiteness properties of groups, Invent. Math. 129 (1997) 445–470
• G E Bredon, Equivariant cohomology theories, Lecture Notes in Mathematics 34, Springer, Berlin (1967)
• M R Bridson, Finiteness properties for subgroups of ${\rm GL}(n,\bf Z)$, Math. Ann. 317 (2000) 629–633
• K S Brown, Presentations for groups acting on simply-connected complexes, J. Pure Appl. Algebra 32 (1984) 1–10
• K S Brown, Finiteness properties of groups, from: “Proceedings of the Northwestern conference on cohomology of groups”, J. Pure Appl. Algebra 44 (1987) 45–75
• F Bruhat, J Tits, Schémas en groupes et immeubles des groupes classiques sur un corps local, Bull. Soc. Math. France 112 (1984) 259–301
• K-U Bux, K Wortman, Connectivity properties of horospheres in Euclidean buildings and applications to finiteness properties of discrete groups, Invent. Math. 185 (2011) 395–419
• M Fluch, S Witzel, Brown's criterion in Bredon homology, to appear in Homology, Homotopy Appl.
• D R Grayson, Finite generation of $K$-groups of a curve over a finite field (after Daniel Quillen), from: “Algebraic $K$-theory, Part I”, (R K Dennis, editor), Lecture Notes in Math. 966, Springer, Berlin (1982) 69–90
• F Grunewald, V P Platonov, On finite extensions of arithmetic groups, C. R. Acad. Sci. Paris Sér. I Math. 325 (1997) 1153–1158
• D H Kochloukova, C Martínez-Pérez, B E A Nucinkis, Centralisers of finite subgroups in soluble groups of type ${\rm FP}\sb n$, Forum Math. 23 (2011) 99–115
• P H Kropholler, C Martinez-Pérez, B E A Nucinkis, Cohomological finiteness conditions for elementary amenable groups, J. Reine Angew. Math. 637 (2009) 49–62
• I J Leary, B E A Nucinkis, Some groups of type $VF$, Invent. Math. 151 (2003) 135–165
• W Lück, Transformation groups and algebraic $K$–theory, Lecture Notes in Mathematics 1408, Springer, Berlin (1989)
• W Lück, D Meintrup, On the universal space for group actions with compact isotropy, from: “Geometry and topology: Aarhus”, (K Grove, I H Madsen, E K Pedersen, editors), Contemp. Math. 258, Amer. Math. Soc. (2000) 293–305
• V P Platonov, The theory of algebraic linear groups and periodic groups, Izv. Akad. Nauk SSSR Ser. Mat. 30 (1966) 573–620 In Russian; translated in Am. Math. Soc. Transl. II. Ser. 69 (1968) 61–110
• G Prasad, J-K Yu, On finite group actions on reductive groups and buildings, Invent. Math. 147 (2002) 545–560
• M Ronan, Lectures on buildings, Perspectives in Mathematics 7, Academic Press, Boston, MA (1989)
• B Schulz, Spherical subcomplexes of spherical buildings, Geom. Topol. 17 (2013) 531–562
• R Strebel, Finitely presented soluble groups, from: “Group theory”, (K W Gruenberg, J E Roseblade, editors), Academic Press, London (1984) 257–314
• J Tits, Reductive groups over local fields, from: “Automorphic forms, representations and $L$–functions, Part 1”, (A Borel, W Casselman, editors), Proc. Sympos. Pure Math. 33, Amer. Math. Soc. (1979) 29–69