The Lagarias-Wang finiteness conjecture is false

You need to know: Matrix, square matrix, (complex) eigenvalue of a square matrix, absolute value of a complex number, $\limsup$ .

Background: The spectral radius $\rho(A)$ of a square matrix A is the largest absolute value of an eigenvalue of A. Let $\Sigma$ be a finite set of $n \times n$ matrices. Let $\rho_k(\Sigma)=\max\{\rho(A_1 A_2 \dots A_k)^{1/k}, \, A_i \in \Sigma, \, i=1,2,\dots,k\}$ . The quantity $\rho(\Sigma) = \limsup\limits_{k\to\infty} \rho_k(\Sigma)$ is called the generalized spectral radius of $\Sigma$ .

The Theorem: On 11th July 2000, Thierry Bousch and Jean Mairesse submitted to the Journal of the AMS a paper in which they proved, among other results, the existence of a finite set $\Sigma$ of matrices (in fact, $\Sigma$ can consist of two $2 \times 2$ matrices) such that $\rho(\Sigma) > \rho_k(\Sigma)$ for all $k \geq 0$ .

Short context: The generalized spectral radius is an important concept useful in a wide range of contexts. It is known that $\rho(\Sigma) \geq \rho_k(\Sigma)$ for all $k \geq 0$ . In 1995, Lagarias and Wang conjectured that, for every $\Sigma$ , there is a k such that $\rho(\Sigma) = \rho_k(\Sigma)$ . This statement became known as the finiteness conjecture. The Theorem disproves this conjecture.