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Register a new userMinimal exponential growth rates of metabelian Baumslag-Solitar groups and lamplighter groups
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We prove that for any prime $pgeq 3$ the minimal exponential growth rate of the Baumslag-Solitar group $BS(1,p)$ and the lamplighter group $mathcal{L}_p=(mathbb{Z}/pmathbb{Z})wr mathbb{Z}$ 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 $mathcal{L}_2$ is equal to the golden ratio $(1+sqrt5)/2$. The latter value also serves to show that the lower bound of A.Mann from [Mann, Journal of Algebra 326, no. 1 (2011) 208--217] for the growth rates of non-semidirect HNN extensions is optimal.
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In this paper we classify Baumslag-Solitar groups up to commensurability. In order to prove our main result we give a solution to the isomorphism problem for a subclass of Generalised Baumslag-Solitar groups.
We study convergent sequences of Baumslag-Solitar groups in the space of marked groups. We prove that BS(m,n) --> F_2 for |m|,|n| --> infty and BS(1,n) --> Z wr Z for |n| --> infty. For m fixed, |m|>1, we show that the sequence (BS(m,n))_n is not convergent and characterize many convergent subsequences. Moreover if X_m is the set of BS(m,n)s for n relatively prime to m and |n|>1, then the map BS(m,n) mapsto n extends continuously on the closure of X_m to a surjection onto invertible m-adic integers.
In this paper we give asymptotics for the conjugacy growth of the soluble Baumslag-Solitar groups $BS(1,k)$, $kgeq 2$, with respect to the standard generating set, by providing a complete description of geodesic conjugacy representatives. We show that the conjugacy growth series for these groups are transcendental, and give formulas for the series. As a result of our computation we also establish that in each $BS(1,k)$ the conjugacy and standard growth rates are equal.
For an element in $BS(1,n) = langle t,a | tat^{-1} = a^n rangle$ written in the normal form $t^{-u}a^vt^w$ with $u,w geq 0$ and $v in mathbb{Z}$, we exhibit a geodesic word representing the element and give a formula for its word length with respect to the generating set ${t,a}$. Using this word length formula, we prove that there are sets of elements of positive density of positive, negative and zero conjugation curvature, as defined by Bar Natan, Duchin and Kropholler.
We exhibit a regular language of geodesics for a large set of elements of $BS(1,n)$ and show that the growth rate of this language is the growth rate of the group. This provides a straightforward calculation of the growth rate of $BS(1,n)$, which was initially computed by Collins, Edjvet and Gill in [5]. Our methods are based on those we develop in [8] to show that $BS(1,n)$ has a positive density of elements of positive, negative and zero conjugation curvature, as introduced by Bar-Natan, Duchin and Kropholler in [1].
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