The Distant Graph of the Ring of Integers and Its Representations in the Modular Group

There is presented an infinite class of subgroups of the modular group \(\mathrm{PSL}(2,\mathbb {Z})\) that serve as Cayley representations of the distant graph of the projective line of integers. They are infinite countable free products of subgroups of \(\mathrm{PSL}(2,\mathbb {Z})\) isomorphic with \(\mathbb {Z}_2\), \(\mathbb {Z}_3\) and \(\mathbb {Z}\) subject to the restriction that the number of copies of \(\mathbb {Z}\) is 0 or 2. The proof technique is based on a 1–1 correspondence between some involutions \(\iota \) of \(\mathbb {Z}\) that fulfill the equation

and groups from this class.

Authors and Affiliations

1. Faculty of Mathematics and Computer Science, University of Warmia and Mazury in Olsztyn, Słoneczna 54, 10-790, Olsztyn, Poland

   Andrzej Matraś & Artur Siemaszko

Corresponding author

Correspondence to Artur Siemaszko.

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