#### Vol. 7, No. 5, 2013

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Weakly commensurable $S$-arithmetic subgroups in almost simple algebraic groups of types $\mathsf{B}$ and $\mathsf{C}$

### Skip Garibaldi and Andrei Rapinchuk

Vol. 7 (2013), No. 5, 1147–1178
##### Abstract

Let ${G}_{1}$ and ${G}_{2}$ be absolutely almost simple algebraic groups of types ${B}_{\ell }$ and ${C}_{\ell }$, respectively, defined over a number field $K$. We determine when ${G}_{1}$ and ${G}_{2}$ have the same isomorphism or isogeny classes of maximal $K$-tori. This leads to the necessary and sufficient conditions for two Zariski-dense $S$-arithmetic subgroups of ${G}_{1}$ and ${G}_{2}$ to be weakly commensurable.

##### Keywords
isogenous maximal tori, weak commensurability, isomorphic maximal tori
##### Mathematical Subject Classification 2010
Primary: 20G15
Secondary: 11E57, 14L35, 20G30
##### Milestones
Received: 20 January 2012
Revised: 29 April 2012
Accepted: 7 June 2012
Published: 6 September 2013
##### Authors
 Skip Garibaldi Institute for Pure and Applied Mathematics 460 Portola Plaza Box 957121 Los Angeles, CA 90095-7121 United States http://www.mathcs.emory.edu/~skip/ Andrei Rapinchuk Department of Mathematics University of Virginia Charlottesville, VA 22904 United States http://www.math.virginia.edu/Faculty/Rapinchuk/