2026/07/22 by Mohammad F. Marashdeh
#math.GR #math.OA
Let G be a finitely generated group with a surjective homomorphism π\colon G→\Z. The directional growth spectrum I(β) is the exponential rate of the number of elements of length at most n lying over \lfloorβn\rfloor; its maximum is the growth rate. Wherever I has been computed---free, hyperbolic and relatively hyperbolic groups, free abelian groups---it is strictly concave and real-analytic, as Perron--Frobenius theory dictates. We compute I in closed form for the lamplighter groups F\wr\Z, |F|=r+1, with the standard generators, and obtain a different picture: I is affine on [-β*,β*] and strictly concave beyond it, with a second-order transition at β*. Elements conditioned to the affine phase backtrack macroscopically, with lamp density independent of β. The series ∑x s|x|yπ(x) is rational, and the transition is an exchange of dominant singularities, the inner one independent of y. The peak of I is logω with ω2=ω+r, the central value logρ with ρ3=ρ+r; the base group is therefore co-amenable yet grows strictly slower, by an amount unbounded in r. For r=1 these constants are the golden ratio and the plastic number.