We classify the growth of a k-regular sequence based on information from its k-kernel. In order to provide such a classification, we introduce the notion of a growth exponent for k-regular sequences and show that this exponent is equal to the joint spectral radius of any set of a special class of matrices determined by the k-kernel.

We recast the notion of joint spectral radius in setting groups acting by isometries on non-positively curved spaces and give geometric versions results Berger–Wang Bochi valid for δ-hyperbolic symmetric non-compact type. This method produces nice hyperbolic elements many classical settings. Applications to uniform growth are given, particular a new proof generalization theorem Besson–Courtois–...

The problem of computation of the joint (generalized) spectral radius of matrix sets has been discussed in a number of publications. In the paper an iteration procedure is considered that allows to build numerically Barabanov norms for the irreducible matrix sets and simultaneously to compute the joint spectral radius of these sets.

It is proved that the Green’s function of the simple random walk on a surface group of large genus decays exponentially in distance at the (inverse) spectral radius. It is also shown that Ancona’s inequalities extend to the spectral radius R, and therefore that the Martin boundary for R−potentials coincides with the natural geometric boundary S. This implies that the Green’s function obeys a po...