Narrow Results

Motivated by results about “untangling” closed curves on hyperbolic surfaces, Gupta and Kapovich introduced the primitivity and simplicity index functions for finitely generated free groups, $d_{\mathrm{\text{prim}}}(g;F_N)$ and $d_{\mathrm{\text{simp}}}(g;F_N)$, where $1\ne g\in F_N$, and obtained some upper and lower bounds for these functions. In this paper, we study the behavior of the sequence $d_{\mathrm{\text{prim}}}(a^n{b}^n;F(a,b))$ as $n\to \infty$. Answering a question from [17], we prove that this sequence is unbounded and that for $n_i=\mathrm{\text{lcm}}(1,2,\dots ,i)$, we have $|d_{\mathrm{\text{prim}}}(a^{n_i}{b}^{n_i};F(a,b))-log(n_i)|=o(log(n_i))$. By contrast, we show that for all $n\ge 2$, one has $d_{\mathrm{\text{simp}}}(a^n{b}^n;F(a,b))=2$. In addition to topological and group-theoretic arguments, number-theoretic considerations, particularly the use of asymptotic properties of the second Chebyshev function, turn out to play a key role in the proofs.

We provide an algorithm that, given a finite set of generators for a subgroup H of a finitely generated free group F, determines whether H is echelon or not and, in case of affirmative answer, also computes a basis with respect to which H is in echelon form. This gives an answer to a question of Rosenmann. We also prove, by means of a counterexample, that intersection of two echelon subgroups needs not to be echelon, answering another question of Rosenmann.

We show that any nonabelian free group F of finite rank is homogeneous; that is for any tuples , having the same complete n-type, there exists an automorphism of F which sends ā to .

We further study existential types and show that for any tuples , if ā and have the same existential n-type, then either ā has the same existential type as the power of a primitive element or there exists an existentially closed subgroup E(ā) (respectively ) of F containing ā (respectively ) and an isomorphism with .

We will deal with non-free two-generated torsion-free hyperbolic groups and we show that they are ∃-homogeneous and prime. In particular, this gives concrete examples of finitely generated groups which are prime and not quasi axiomatizable, giving an answer to a question of A. Nies.