7 papers · 1 filter
Automorphic orbits in free groups: recent progress
Vladimir Shpilrain
In this survey, we describe recent progress on asymptotic properties of various automorphic orbits in free groups. In particular, we address the problem of counting potentially pos…
Easy estimates of Lyapunov exponents for random products of matrices
Nadya Nabahi, Vladimir Shpilrain
The problems that we consider in this paper are as follows. Let be square matrices (over reals). Let be a random product of matrices.…
Orbit-blocking words in free groups
Lucy Koch-Hyde, Siobhan O'Connor, Eamonn Olive +1
By strengthening known results about primitivity-blocking words in free groups, we prove that for any nontrivial element w of a free group of finite rank, there are words that cann…
Growth in products of matrices: fastest, average, and generic
Vladimir Shpilrain
The problems that we consider in this paper are as follows. Let A and B be 2x2 matrices (over reals). Let w(A, B) be a word of length n. After evaluating w(A, B) as a product of ma…
Linear average-case complexity of algorithmic problems in groups
Alexander Olshanskii, Vladimir Shpilrain
The worst-case complexity of group-theoretic algorithms has been studied for a long time. Generic-case complexity, or complexity on random inputs, was introduced and studied relati…
Orbit-blocking words and the average-case complexity of Whitehead's problem in the free group of rank 2
Lucy Hyde, Siobhan O'Connor, Vladimir Shpilrain
Let F_2 denote the free group of rank 2. Our main technical result of independent interest is: for any element u of F_2, there is g in F_2 such that no cyclically reduced image of…