We prove that every mapping torus of any free group endomorphism is residually finite. We show how to use a not yet published result of E. Hrushovski to extend our result to arbitrary linear groups. The proof uses algebraic self-maps of affine spaces over finite fields. In particular, we prove that when such a map is dominant, the set of its fixed closed scheme points is Zariski dense in the affine space.

A sequence of functions F = {f n (q)} ∞ n=1 satisfies the functional equation for multiplication of quantum integers if f mn (q) = f m (q)f n (q m ) for all positive integers m and n. This paper describes the structure of all sequences of rational functions with coefficients in Q that satisfy this functional equation.

The two-dimensional case of the famous Jacobian conjecture of O.-H. Keller asserts that every unramified polynomial self-map of an affine plane is invertible. Many geometric approaches to this conjecture involve divisorial valuations of the field C(x, y), centered outside of the affine plane. Two integer invariants of these valuations naturally appear in this context. In this paper we study these invariants using combinatorics of weighted graphs. In particular, we prove that whenever both invariants are fixed, the corresponding valuations form a finite number of families up to plane automorphisms.

Abstract. We give an almost complete classification of empty lattice simplices in dimension 4 using the conjectural results of Mori-Morrison-Morrison, which were later proved by Sankaran and Bober. In particular, all of these simplices correspond to cyclic quotient singularities, and all but finitely many of them have width bounded by 2.

It was conjectured by M c Kernan and Shokurov that for all Mori contractions from X to Y of given dimensions, for any positive ε there is a positive δ such that if X is ε-log terminal, then Y is δ-log terminal. We prove this conjecture in the toric case and discuss the dependence of δ on ε, which seems mysterious.