We show that the elementary theory of the free group – and more generally the elementary theory of any torsion-free hyperbolic group – is n-ample for any n ≥ 1. We also give an explicit description of the imaginary algebraic closure in free groups.

RKMK methods and Crouch-Grossman methods are two classes of Lie group methods. The former is using flows and commutators of a Lie algebra of vector fields as a part of the method definition. The latter uses only compositions of flows of such vector fields, but the number of flows which needs to be computed is much higher than in the RKMK methods. We present a new type of methods which avoids th...

Group signatures are a useful cryptographic construct for privacy-preserving non-repudiable authentication, and there have been many group signature schemes. In this paper, we introduce a variant of group signatures that offers two new security properties called leak-freedom and immediate-revocation. Intuitively, the former ensures that an insider (i.e., an authorized but malicious signer) be u...

We present a coarse convexity result for the dynamics of free group automorphisms: Given an automorphism φ of a finitely generated free group F , we show that for all x ∈ F and 0 ≤ i ≤ N , the length of φ(x) is bounded above by a constant multiple of the sum of the lengths of x and φ (x), with the constant depending only on φ.

We study model-theoretic and stability-theoretic properties of the nonabelian free group in the light of Sela’s recent result [15] on stability and results announced by Bestvina and Feighn on “negligible subsets” of free groups. We point out analogies between the free group and socalled bad groups of finite Morley rank, and prove “non CM -triviality” of the free group.