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Register a new userThe conjugacy growth of the soluble Baumslag-Solitar groups
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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.
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A generalized Baumslag-Solitar group is the fundamental group of a graph of groups all of whose vertex and edge groups are infinite cyclic. Levitt proves that any generalized Baumslag-Solitar group has property R-infinity, that is, any automorphism has an infinite number of twisted conjugacy classes. We show that any group quasi-isometric to a generalized Baumslag-Solitar group also has property R-infinity. This extends work of the authors proving that any group quasi-isometric to a solvable Baumslag-Solitar BS(1,n) group has property R-infinity, and relies on the classification of generalized Baumslag-Solitar groups given by Whyte.
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.
We say that a group has property $R_{infty}$ if any group automorphism has an infinite number of twisted conjugacy classes. Felshtyn and Goncalves prove that the solvable Baumslag-Solitar groups BS(1,m) have property $R_{infty}$. We define a solvable generalization $Gamma(S)$ of these groups which we show to have property $R_{infty}$. We then show that property $R_{infty}$ is geometric for these groups, that is, any group quasi-isometric to $Gamma(S)$ has property $R_{infty}$ as well.
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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