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The Farrell-Tate and Bredon homology for $$\operatorname{PSL}_4(\mathbb{Z})$$ via cell subdivisions. (English) Zbl 1428.11097
There are two competing notions of cellular group action on a CW-complex $$X$$. The weaker notion imposes that the group action respects the cell structure of $$X$$. The stronger and more common notion includes the additional assumption that if a group element stabilizes a cell, then it stabilizes the cell pointwise. In the article under review the latter actions are called rigid. The authors present subdivision algorithms which allow to refine a cell complex with a (weakly) cellular action into a rigid one, with the aim of avoiding the creation of many new cells.
The research is motivated by computations of the cohomology of arithmetic groups. To compute the homology of an arithmetic group $$\Gamma$$ it is usually sufficient to have a cell structure on the associated symmetric space such that the action of $$\Gamma$$ is (weakly) cellular. The authors argue that the computation of Bredon homology and Farrell-Tate homology requires rigid cell structures. To this end they describe three algorithms which produces suitable subdivisons and briefly discuss their advantages and disadvantages.
The methods are applied to compute some examples. The authors discuss the Farrell-Tate cohomology modulo $$2$$ of $$\mathrm{SL}_3(\mathbb{Z})$$, and the Farrell-Tate cohomology modulo $$3$$ and $$5$$ of $$\mathrm{Sp}_4(\mathbb{Z})$$. In particular, they describe the Bredon homology and the Farrel-Tate cohomology modulo $$3$$ and $$5$$ of $$\mathrm{PSL}_4(\mathbb{Z})$$.
##### MSC:
 11F75 Cohomology of arithmetic groups
##### Keywords:
cohomology of arithmetic groups; computational methods
GAP; HAP; PFPK
Full Text:
##### References:
 [1] Braun, Oliver; Coulangeon, Renaud; Nebe, Gabriele; Schönnenbeck, Sebastian, Computing in arithmetic groups with Voronoï’s algorithm, J. Algebra, 435, 263-285, (2015), Zbl 1323.16014 · Zbl 1323.16014 [2] Brown, Kenneth S., Cohomology of Groups, Graduate Texts in Mathematics, vol. 87, (1994), Springer-Verlag: Springer-Verlag New York, corrected reprint of the 1982 original, MR1324339 [3] Brownstein, Alan; Lee, Ronnie, Cohomology of the symplectic group $$\operatorname{Sp}_4(\mathbf{Z})$$. I. The odd torsion case, Trans. Am. Math. Soc., 334, 2, 575-596, (1992) · Zbl 0762.11022 [4] Bui, Anh Tuan; Ellis, Graham J., Computing Bredon homology of groups, J. Homotopy Relat. Struct., 11, 4, 715-734, (2016), MR3578995 · Zbl 1400.20052 [5] Bui, Anh Tuan; Rahm, Alexander D., Torsion subcomplexes subpackage, version 2.1, accepted sub-package in HAP (Homological Algebra Programming) in the computer algebra system GAP, 2018, source code available at [6] Dutour Sikirić, Mathieu; Ellis, Graham J.; Schuermann, Achill, On the integral homology of $$\operatorname{PSL}_4(\mathbb{Z})$$ and other arithmetic groups, J. Number Theory, 131, 12, 2368-2375, (2011), Zbl 1255.11028 · Zbl 1255.11028 [7] Dutour Sikirić, Mathieu; Gangl, Herbert; Gunnells, Paul E.; Hanke, Jonathan; Schuermann, Achill; Yasaki, Dan, On the cohomology of linear groups over imaginary quadratic fields, J. Pure Appl. Algebra, 220, 7, 2564-2589, (2016) · Zbl 1408.11051 [8] Ellis, Graham, Homological algebra programming, (Computational Group Theory and the Theory of Groups, (2008)), 63-74, MR2478414 (2009k:20001), implemented in the HAP package in the GAP computer algebra system · Zbl 1155.20051 [9] Elbaz-Vincent, Philippe; Gangl, Herbert; Soulé, Christophe, Perfect forms, K-theory and the cohomology of modular groups, Adv. Math., 245, 587-624, (2013), Zbl 1290.11104 · Zbl 1290.11104 [10] Ellis, Graham; Hegarty, Fintan, Computational homotopy of finite regular CW-spaces, J. Homotopy Relat. Struct., 9, 1, 25-54, (2014), Zbl 1311.55008 · Zbl 1311.55008 [11] MacPherson, Robert; McConnell, Mark, Explicit reduction theory for Siegel modular threefolds, Invent. Math., 111, 3, 575-625, (1993), MR1202137 · Zbl 0789.11029 [12] Mislin, Guido; Valette, Alain, Proper Group Actions and the Baum-Connes Conjecture, Advanced Courses in Mathematics. CRM Barcelona, (2003), Birkhäuser Verlag: Birkhäuser Verlag Basel, MR2027168 (2005d:19007), Zbl 1028.46001 · Zbl 1028.46001 [13] Rahm, Alexander D., Accessing the cohomology of discrete groups above their virtual cohomological dimension, J. Algebra, 404, 152-175, (2014), MR3177890 · Zbl 1296.11050 [14] Rahm, Alexander D., On the equivariant K-homology of $$\operatorname{PSL}_2$$ of the imaginary quadratic integers, Ann. Inst. Fourier, 66, 4, 1667-1689, (2016) · Zbl 1360.55007 [15] Rahm, Alexander D.; Wendt, Matthias, On Farrell-Tate cohomology of $$\operatorname{SL}_2$$ over S-integers, J. Algebra, 512, 427-464, (2018), MR3841530 · Zbl 1447.20017 [16] Reiner, Irving, Integral representations of cyclic groups of prime order, Proc. Am. Math. Soc., 8, 142-146, (1957), MR0083493 · Zbl 0077.25103 [17] Sánchez-García, Rubén J., Bredon homology and equivariant K-homology of $$\operatorname{SL}(3, \mathbb{Z})$$, J. Pure Appl. Algebra, 212, 5, 1046-1059, (2008), MR2387584 (2009b:19007) · Zbl 1145.19005 [18] Sánchez-García, Rubén J., Equivariant K-homology for some Coxeter groups, J. Lond. Math. Soc. (2), 75, 3, 773-790, (2007), MR2352735 (2009b:19006) · Zbl 1175.19004 [19] Schönnenbeck, Sebastian, Resolutions for unit groups of orders, J. Homotopy Relat. Struct., 12, 4, 837-852, (2017), MR3723461 · Zbl 1381.16017 [20] Serre, Jean-Pierre, Trees, Springer Monographs in Mathematics, (2003), Springer-Verlag: Springer-Verlag Berlin, translated from the French original by John Stillwell; corrected 2nd printing of the 1980 English translation, MR1954121 · Zbl 1013.20001 [21] Soulé, Christophe, The cohomology of $$\operatorname{SL}_3(\mathbf{Z})$$, Topology, 17, 1, 1-22, (1978) · Zbl 0382.57026
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