Period sets of linear toral endomorphisms on T2T^2

The period set of a dynamical system is defined as the subset of all integers $n$ such that the system has a periodic orbit of length $n$. Based on known results on the intersection of period sets of torus maps within a homotopy class, we give a complete classification of the period sets of (not necessarily invertible) toral endomorphisms on the $2$--dimensional torus $\mathbb{T}^2$.Comment: 10 page

Lifting Finite Groups of Outer Automorphisms of Free Groups, Surface Groups and their Abelianizations

In the present note, in part written as a survey, we discuss the possibility of lifting finite subgroups, and in particular finite cyclic subgroups, with respect to the canonical projections between automorphism and outer automorphism groups of free groups, surface groups and their abelianizations

VORONOI COMPLEXES IN HIGHER DIMENSIONS, COHOMOLOGY OF GLN(Z)GL_N (Z) FOR N≥8N\ge 8 AND THE TRIVIALITY OF K8(Z)K_8 (Z)

We enumerate the low dimensional cells in the Voronoi cell complexes attached to the modular groups $SL_N (Z)$ and $GL_N (Z)$ for $N = 8, 9, 10, 11$, using quotient sublattices techniques for $N = 8, 9$ and linear programming methods for higher dimensions. These enumerations allow us to compute some cohomology of these groups and prove that $K_8 (Z) = 0$, providing new knowledge on the Kummer-Vandiver conjecture