The famous "Tits alternative" states that a pair of matrices which generate a non-solvable group must contain a free group. Around 2008, Breuillard proved a strong version of this result by showing that a pair of (algebraic) matrices generating a non-solvable group must generate a matrix with large eigenvalues (in some valuation). We will discuss this theorem and some of its applications. We will also offer a new proof based on the existence of curious word maps known as almost laws. Joint work with Homin Lee and Lvzhou Chen.
Speaker(s): Sebastian Hurtado (U Chicago)