Semidualities from products of trees

Studenmund, Daniel¹ ; Wortman, Kevin¹

¹ Department of Mathematics, University of Utah, Salt Lake City, UT, United States

Geometry & topology, Tome 22 (2018) no. 3, pp. 1717-1758.

Résumé

Let $K$ be a global function field of characteristic $p$, and let $\Gamma$ be a finite-index subgroup of an arithmetic group defined with respect to $K$ and such that any torsion element of $\Gamma$ is a $p$–torsion element. We define semiduality groups, and we show that $\Gamma$ is a $\mathbb{Z}\left[1∕p\right]$–semiduality group if $\Gamma$ acts as a lattice on a product of trees. We also give other examples of semiduality groups, including lamplighter groups, Diestel–Leader groups, and countable sums of finite groups.

Détail

DOI : 10.2140/gt.2018.22.1717

Classification : 20G10, 57M07, 57Q05
Keywords: arithmetic groups, cohomology of arithmetic groups, semiduality, lamplighter group, Diestel–Leader groups

Bibliographie

[1] L Bartholdi, M Neuhauser, W Woess, Horocyclic products of trees, J. Eur. Math. Soc. 10 (2008) 771 | DOI

[2] H Behr, Endliche Erzeugbarkeit arithmetischer Gruppen über Funktionenkörpern, Invent. Math. 7 (1969) 1 | DOI

[3] M Bestvina, A Eskin, K Wortman, Filling boundaries of coarse manifolds in semisimple and solvable arithmetic groups, J. Eur. Math. Soc. 15 (2013) 2165 | DOI

[4] R Bieri, B Eckmann, Finiteness properties of duality groups, Comment. Math. Helv. 49 (1974) 74 | DOI

[5] A Borel, J P Serre, Corners and arithmetic groups, Comment. Math. Helv. 48 (1973) 436 | DOI

[6] A Borel, J P Serre, Cohomologie d’immeubles et de groupes S–arithmétiques, Topology 15 (1976) 211 | DOI

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

[8] K S Brown, Finiteness properties of groups, J. Pure Appl. Algebra 44 (1987) 45 | DOI

[9] K U Bux, Finiteness properties of soluble arithmetic groups over global function fields, Geom. Topol. 8 (2004) 611 | DOI

[10] K U Bux, R Köhl, S Witzel, Higher finiteness properties of reductive arithmetic groups in positive characteristic: the rank theorem, Ann. of Math. 177 (2013) 311 | DOI

[11] K U Bux, K Wortman, Finiteness properties of arithmetic groups over function fields, Invent. Math. 167 (2007) 355 | DOI

[12] 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 | DOI

[13] G Gandini, Bounding the homological finiteness length, Bull. Lond. Math. Soc. 44 (2012) 1209 | DOI

[14] G Harder, Minkowskische Reduktionstheorie über Funktionenkörpern, Invent. Math. 7 (1969) 33 | DOI

[15] P H Kropholler, On groups of type (FP)∞, J. Pure Appl. Algebra 90 (1993) 55 | DOI

[16] U Stuhler, Homological properties of certain arithmetic groups in the function field case, Invent. Math. 57 (1980) 263 | DOI

[17] C A Weibel, An introduction to homological algebra, 38, Cambridge Univ. Press (1994) | DOI

[18] K Wortman, A finitely presented solvable group with a small quasi-isometry group, Michigan Math. J. 55 (2007) 3 | DOI