McKay graphs for alternating and classical groups

Let G be a finite group, andαa nontrivial character of G. The McKay graph M (G,α) has the irreducible characters of Gas vertices, with an edge fromχ1toχ2ifχ2is a constituent ofαχ1. We study the diameters of McKay graphs for finite simple groups G. For alternating groups G=An, we prove a conjecture made in [20]: there is an absolute constant C such that diam M (G,α)≤ C log | G| log α (1)for all nontrivial irreducible characters α of G. Also for classical groups of symplectic or orthogonal type of rank r, we establish a linear upper bound Cr on the diameters of all nontrivial McKay graphs. Finally, we provide some sufficient conditions for a productχ1χ2···χlof irreducible characters of some finite simple groups G to contain all irreducible characters of G as constituents

Alpha-invariants and purely log terminal blow-ups

We prove that the sum of the $\alpha$-invariants of two different Koll\'ar components of a Kawamata log terminal singularity is less than $1$.Comment: 12 page

Character ratios for exceptional groups of Lie type

We prove character ratio bounds for finite exceptional groups G(q) of Lie type. These take the form |χ(g)|χ(1)≤cqk for all nontrivial irreducible characters χ and nonidentity elements g⁠, where c is an absolute constant, and k is a positive integer. Applications are given to bounding mixing times for random walks on these groups and also diameters of their McKay graphs

On the diameters of McKay graphs for finite simple groups

Let Gbe a finite group, andαa nontrivial character of G. The McKay graph M(G,α) has the irreducible characters of Gas vertices, with an edge from χ1 to χ2 if χ2 is a constituent of αχ1. We study the diameters of McKay graphs for simple groups G of Lie type. We show that for anyα, the diameter is bounded by a quadratic function of the rank, and obtain much stronger bounds for G= PSLn(q) or PSUn(q

Character ratios, representation varieties and random generation of finite groups of Lie type

We use character theory of finite groups of Lie type to establish new results on representation varieties of Fuchsian groups, and also on probabilistic generation of groups of Lie type.Comment: 32 pages, to appear in Advances in Mathematic