Asymptotic density in free groups and Z k, visible points and test elements (original) (raw)

DENSITIES IN FREE GROUPS AND Z k , VISIBLE POINTS AND TEST ELEMENTS

2005

In this article we relate two different densities. Let Fk be the free group of finite rank k ≥ 2 and let α be the abelianization map from Fk onto Zk. We prove that if S ⊆ Zk is invariant under the natural action of SL(k, Z) then the asymptotic density of S in Zk and the annular density of its full preimage α−1(S) in Fk are equal. This implies, in particular, that for every integer t ≥ 1, the annular density of the set of elements in Fk that map to t-th powers of primitive elements in Zk is equal to 1 tkζ(k) , where ζ is the Riemann zeta-function. An element g of a group G is called a test element if every endomorphism of G which fixes g is an automorphism of G. As an application of the result above we prove that the annular density of the set of all test elements in the free group F (a, b) of rank two is 1 − 6 π2 . Equivalently, this shows that the union of all proper retracts in F (a, b) has annular density 6 π2 . Thus being a test element in F (a, b) is an “intermediate property” ...

Densities in free groups and mathbbZk\mathbb{Z}^kmathbbZk, Visible Points and Test Elements

Mathematical Research Letters, 2007

In this article we relate two different densities. Let F k be the free group of finite rank k ≥ 2 and let α be the abelianization map from F k onto Z k . We prove that if S ⊆ Z k is invariant under the natural action of SL(k, Z) then the asymptotic density of S in Z k and the annular density of its full preimage α −1 (S) in F k are equal. This implies, in particular, that for every integer t ≥ 1, the annular density of the set of elements in F k that map to t-th powers of primitive elements in Z k is equal to to 1 t k ζ(k) , where ζ is the Riemann zeta-function.

1 Growth in Free Groups (And Other Stories)–Twelve Years Later

2016

We start by studying the distribution of (cyclically reduced) elements of the free groups F n with respect to their abelianization (or equivalently, their class in H 1 (F n , Z)). We derive an explicit generating function, and a limiting distribution, by means of certain results (of independent interest) on Chebyshev polynomials; we also prove that the reductions mod p (p-an arbitrary prime) of these classes are asymptotically equidistributed, and we study the deviation from equidistribution. We extend our techniques to a more general setting and use them to study the statistical properties of long cycles (and paths) on regular (directed and undirected) graphs. We return to the free group to study some growth functions of the number of conjugacy classes as a function of their cyclically reduced length.

Growth in free groups (and other stories)—twelve years later

Illinois Journal of Mathematics, 2010

We start by studying the distribution of (cyclically reduced) elements of the free groups Fn with respect to their Abelianization (or equivalently, their class in H1(Fn, Z)). We derive an explicit generating function, and a limiting distribution, by means of certain results (of independent interest) on Chebyshev polynomials; we also prove that the reductions mod p (pan arbitrary prime) of these classes are asymptotically equidistributed, and we study the deviation from equidistribution. We extend our techniques to a more general setting and use them to study the statistical properties of long cycles (and paths) on regular (directed and undirected) graphs. We return to the free group to study some growth functions of the number of conjugacy classes as a function of their cyclically reduced length.

Growth in free groups (and other stories

1999

We start by studying the distribution of (cyclically reduced) elements of the free groups with respect to their abelianization. We derive an explicit generating function, and a limiting distribution, by means of certain results (of independent interest) on Chebyshev polynomials; we also prove that the reductions modp\mod pmodp ($p$ -- an arbitrary prime) of these classes are asymptotically equidistributed, and we study the deviation from equidistribution. We extend our techniques to a more general setting and use them to study the statistical properties of long cycles (and paths) on regular (directed and undirected) graphs. We return to the free group to study some growth functions of the number of conjugacy classes as a function of their cyclically reduced length.

Residual properties of free groups II

Bulletin of the Australian Mathematical Society, 1972

In this paper it is proved that non-abelian free groups are residually [x, y \ x =1, y n =l, x = y) if and only if min{(m, k), (n, h)} is greater than 1 , and not both of (m, k) and (n, h) are 2 (where 0 is taken as greater than any natural number). The proof makes use of a result, possibly of independent interest, concerning the existence of certain automorphisms of the free group of rank two. A useful criterion which enables one to prove that non-abelian free groups are residually G for a large number of groups G is also given.