Property tau Need Not Survive Intersections of Normal Subgroups

We construct a family of finite-index normal subgroups of a fixed finitely generated free group which has property \((\tau)\), whereas its prefix-intersection chain does not. The construction uses Kassabov's bounded-degree alternating-group expanders. Two quotient maps have identical expanding generators and send one additional generator to, respectively, the identity and a \(3\)-cycle. Their joint image is the full product \(\operatorname{Alt}(N)\times\operatorname{Alt}(N)\). In its action on \(N^2\) ordered pairs, the diagonal provides a mean-zero test function with lazy Rayleigh quotient \(3/[2r(N-1)]\), where \(r\) is the fixed number of generators. Pullback transfers this bound to the regular quotient and then to the prefix chain. Thus failure already occurs for a sequence of pairwise intersections, giving a negative answer to Question 9 in Abért's list.