ARITHMETIC GROUPS OVER FUNCTION-FIELDS I - A COMPLETE CHARACTERIZATION OF FINITELY GENERATED AND FINITELY PRESENTED ARITHMETIC SUBGROUPS OFREDUCTIVE ALGEBRAIC-GROUPS

Arithmetic subgroups of reductive algebraic groups over (global) funct ion fields are eventually finally generated and finitely presented. Bu t in contrast to the number field case there exist exceptions for low ranks of the group and small arithmetic rings. The precise conditions are not independent: the sum of local ranks over the defining set of p rimes is decisive.