Abstract
Abstract The study of the poset of hyperbolic structures on a group G was initiated by C. Abbott, S. Balasubramanya and D. Osin, [Hyperbolic structures on groups, Algebr. Geom. Topol. 19 2019, 4, 1747–1835, 10.2140/agt.2019.19.1747]. However, this poset is still very far from being understood, and several questions remain unanswered. In this paper, we give a complete description of the poset of hyperbolic structures on the lamplighter groups wr ℤ n ℤ {\mathbb{Z}_{n}\mathbin{\mathrm{wr}}\mathbb{Z}} and obtain some partial results about more general wreath products. As a consequence of this result, we answer two open questions regarding quasi-parabolic structures: we give an example of a group G with an uncountable chain of quasi-parabolic structures and prove that the lamplighter groups wr ℤ n ℤ {\mathbb{Z}_{n}\mathbin{\mathrm{wr}}\mathbb{Z}} all have finitely many quasi-parabolic structures.
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