Models of arithmetic and categories with finiteness conditions

Finiteness Obstructions and Euler Characteristics of Categories

We introduce notions of finiteness obstruction, Euler characteristic, L2-Euler characteristic, and Möbius inversion for wide classes of categories. The finiteness obstruction of a category Γ of type (FP) is a class in the projective class group K0(RΓ); the Euler characteristic and L2-Euler characteristic are respectively its RΓ-rank and L2-rank. We also extend the second author’s K-theoretic Mö...

Finiteness for Arithmetic Fewnomial Systems

Suppose L is any finite algebraic extension of the ordinary rational numbers, or any unramified algebraic extension of the p-adic rational numbers. Also let f1, . . . , fk be polynomials in n variables over L such that the total number of monomial terms appearing in at least one fi is exactly m. We prove that the maximum number of isolated roots of F := (f1, . . . , fk) in L n is finite and dep...

Finiteness for Arithmetic Fewnomial Systems

Finiteness of arithmetic Kleinian reflection groups

We prove that there are only finitely many arithmetic Kleinian maximal reflection groups. Mathematics Subject Classification (2000). Primary 30F40; Secondary 57M.

Finiteness of Arithmetic Hyperbolic Reflection Groups

We prove that there are only finitely many conjugacy classes of arithmetic maximal hyperbolic reflection groups.