COUNTING PRIMITIVE ELEMENTS IN FREE GROUPS

Counting Primitive Elements in Free Groups

In this paper it is proved that the set of primitive elements of a nonabelian free group has density zero, i.e. the ratio of primitive elements in increasingly large balls is arbitrarily small. Two notions of density (natural and exponential density) are defined and some of their properties are studied. A class of subsets of the free group (graphical sets) is defined restricting the occurrence ...

Primitive elements for p-divisible groups

We introduce the notion of primitive elements in arbitrary truncated p-divisible groups. By design, the scheme of primitive elements is finite and locally free over the base. Primitive elements generalize the “points of exact order N ,” developed by Drinfeld and Katz-Mazur for elliptic curves.

Generalized Primitive Elements of a Free Group

We study endomorphisms of a free group of finite rank by means of their action on specific sets of elements. In particular, we prove that every endomorphism of the free group of rank 2 which preserves an automorphic orbit (i.e., acts “like an automorphism” on one particular orbit), is itself an automorphism. Then, we consider elements of a different nature, defined by means of homological prope...

Regular sets and counting in free groups

Counting Sum-free Sets in Abelian Groups

In this paper we study sum-free sets of order m in finite Abelian groups. We prove a general theorem about independent sets in 3-uniform hypergraphs, which allows us to deduce structural results in the sparse setting from stability results in the dense setting. As a consequence, we determine the typical structure and asymptotic number of sum-free sets of order m in Abelian groups G whose order ...