Minimal exponential growth rates of metabelian Baumslag–Solitar groups and lamplighter groups

Abstract

Abstract. We prove that for any prime p ≥ 3 the minimal exponential growth rate of the Baumslag-Solitar group BS(1, p) and the lamplighter group Lp = (Z/pZ) ≀ Z are equal. We also show that for p = 2 this claim is not true and the growth rate of BS(1, 2) is equal to the positive root of x 3 −x 2 −2, whilst the one of the lamplighter group L 2 is equal to the golden ratio (1 + √ 5)/2. The latter value also serves to show that the lower bound of A.Mann from [8] for the growth rates of non-semidirect HNN extensions is optimal.

The conjugacy growth of the soluble Baumslag-Solitar groups

“…Suppose w = a x0 ta x1 t⋯a xm−1 t ∈ A m , and take a conjugacy geodesic u of [w]. By Proposition 18, u is a word in A m if it is not excluded by (3) and by the first assertion of this lemma, w is a cyclic permutation of u, thus also a conjugacy geodesic. If u is of the form described in (3), then by the proof above, w = (a r−1 ta r t) m 2 or (a r ta r−1 t) m 2 and has the same length as u, so is also a conjugacy geodesic.…”

“…We start with the odd case. By [8,Theorem (iii)] (see also [3,Lemma 11(b)]) the standard growth rate is the inverse of the smallest absolute value of the real roots of the polynomial 1 − 2t − t 2 + 2t r+2 which appears in the denominator of the standard growth series. But the same polynomial appears in the denominator of and since the smallest absolute value of real roots is ρ o ≤ 1 2 , this will dominate the growth rate of the conjugacy classes in Z k , which is smaller than 2 by Corollary 9.…”

A new proof of the growth rate of the solvable Baumslag–Solitar groups

“…The first computation of this growth rate is due to Collins, Edjvet and Gill in [5] who additionally exhibit the growth series for the group. Bucher and Talambutsa in [3] reprove the results in [5] for prime n; their methods involve understanding the action of the group on its Bass-Serre tree. Their goal is to show that the minimal exponential growth rates of the solvable Baumslag-Solitar group BS(1, n) and the lamplighter group L n = Z n ≀ Z coincide for prime n > 2 but differ for n = 2.…”

“…Their goal is to show that the minimal exponential growth rates of the solvable Baumslag-Solitar group BS(1, n) and the lamplighter group L n = Z n ≀ Z coincide for prime n > 2 but differ for n = 2. In both [3] and [5], as well as in the proofs below, the rate of growth is computed to be the reciprocal of a root of a particular polynomial. When n > 2 is even our polynomials are much simpler than those in [5], and they match those in both references in the remaining cases.…”

Negligibity of elliptic elements in ascending HNN-extensions of $\mathbb{Z}^m$

“…Moreover, the growth series for a 'higher dimensional' generalisation of BS(1, 3) -the ascending HNN-extension of Z m given by the 'cubing homomorphism' u → 3u -has been calculated in [12] with respect to standard generators. Some results on growth of Baumslag-Solitar groups that are independent of the choice of a generating set are also known; in particular, the minimal exponential growth rates (with respect to an arbitrary finite generating set) for BS(1, N ) have been calculated in [1]. This paper aims to provide additional results of this nature.…”