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

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Abstract

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 et al. (AM (Basel) 62:1-11, 1994). Our methods are based on those we develop in Taback and Walker (JTA, to appear) to show that BS(1, n) has a positive density of elements of positive, negative and zero conjugation curvature, as introduced by Bar-Natan et al. (JTA, 2020, https://doi.org/10.1142/S1793525321500096).

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References

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Funding

The first author acknowledges support from Simons Foundation Grant 31736 to Bowdoin College. Both authors thank Moon Duchin, Rob Kropholler and Murray Elder for insightful conversations during the writing of this paper.

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Authors and Affiliations

  1. Department of Mathematics, Bowdoin College, Brunswick, ME, 04011, USA

    Jennifer Taback

  2. Center for Communications Research, La Jolla, CA, 92121, USA

    Alden Walker

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  1. Jennifer Taback
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  2. Alden Walker
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Correspondence to Jennifer Taback.

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Taback, J., Walker, A. A new proof of the growth rate of the solvable Baumslag–Solitar groups. Geom Dedicata 216, 22 (2022). https://doi.org/10.1007/s10711-022-00683-w

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  • DOI: https://doi.org/10.1007/s10711-022-00683-w

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