Direct and Inverse Problems in Baumslag-Solitar Group BS(1,3)

2024/02/25 by Singh, Sandeep, Kaur, Ramandeep
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2402.16128

Abstract

For integers m and n, the Baumslag-Solitar groups, denoted as BS(m,n), are groups generated by two elements with a single defining relation: BS(m,n) = ⟨ a, b | a^mb=baⁿ⟩. The sum of dilates, denoted as r ⋅ A + s ⋅ B for integers r and s, is defined as \ra + sb; a∈ A, b∈ B\. In 2014, Freiman et al. \citefreiman derived direct and inverse results for sums of dilates and applied these findings to address specific direct and inverse problems within Baumslag-Solitar groups, assuming suitable small doubling properties. In 2015, Freiman et al. \citefreiman15 tackled the general problem of small doubling types in a monoid, a subset of the Baumslag-Solitar group BS(1,2). This paper extends these investigations to solve the analogous problem for the Baumslag-Solitar group BS(1,3).

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