The maximal rank of a string group generated by involutions for alternating groups

A string group generated by involutions, or SGGI, is a pair Γ = (G, S), where G is a group and S = {ρ0,..., ρr-1} is an ordered set of involutions generating G and satisfying the following commuting property: for all i, j ϵ {0,..., r - 1 } |i,j| ≠ 1 implies (ρiρj)2 = 1. When S is an independent set, the rank of Γ is the cardinality of S. We determine an upper bound for the rank of an SGGI over the alternating group of degree n. Our bound is tight when n ≡ 0,1,4 (mod 5).