Higher finiteness properties of reductive arithmetic groups in positive characteristic: the rank theorem

Abstract

We show that the finiteness length of an $S$-arithmetic subgroup $\Gamma$ in a noncommutative isotropic absolutely almost simple group $G$ over a global function field is one less than the sum of the local ranks of $G$ taken over the places in $S$. This determines the finiteness properties for arithmetic subgroups in isotropic reductive groups, confirming the conjectured finiteness properties for this class of groups. Our main tool is Behr-Harder reduction theory which we recast in terms of the metric structure of euclidean buildings.

References in corpus (5)

• Bounding the homological finiteness length
• Finiteness Properties of Chevalley Groups over a Polynomial Ring over a Finite Field
• Spherical subcomplexes of spherical buildings
• The sphericity of the Phan geometries of type Bn and Cn and the Phan-type theorem of type F4
• Finiteness Properties of Chevalley Groups over the Laurent Polynomial Ring over a Finite Field

Cited by in corpus (12)

• Stability in the homology of congruence subgroups
• Bounding the homological finiteness length
• On the finiteness length of some soluble linear groups
• Spherical subcomplexes of spherical buildings
• Quotients of the Bruhat-Tits tree by arithmetic subgroups of special unitary groups
• Semidualities from products of trees
• Arithmetic Groups (Banff, Alberta, April 14-19, 2013)
• An Infinite Dimensional Virtual Cohomology Group of ${\text{SL}}_3(\mathbb{Z}[t])$
• An infinitely generated virtual cohomology group for noncocompact arithmetic groups over function fields
• $H^2(S{L}_3(\mathbb{Z}[t]);\mathbb{Q})$ is infinite dimensional
• A building-theoretic approach to relative Tamagawa numbers of quasi-split semisimple groups over global function fields
• On the geometry of global function fields, the Riemann-Roch theorem, and finiteness properties of S-arithmetic groups