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تسجيل مستخدم جديدA new proof of the growth rate of the solvable Baumslag-Solitar groups
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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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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.
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 tha
t 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.
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