For Sierpinski polygon graphs with r not a multiple of 4, isomorphism is classified by a dihedral group action on the defining sequence, and eventually-constant sequences yield exactly two Busemann and countably many non-Busemann horofunctions.
Rationality of the Gromov Boundary of Hyperbolic Groups
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abstract
In [BBM21], Belk, Bleak and Matucci proved that hyperbolic groups can be seen as subgroups of the rational group. In order to do so, they associated a tree of atoms to each hyperbolic group. Not so many connections between this tree and the literature on hyperbolic groups were known. In this paper, we prove an atom-version of the fellow traveler property and exponential divergence, together with other similar results. These leads to several consequences: a bound from above of the topological dimension of the Gromov boundary, the definition of an augmented tree which is quasi-isometric to the Cayley graph and a synchronous recognizer which described the equivalence relation given by the quotient map defined from the end of the tree onto the Gromov boundary.
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Horofunctions of infinite Sierpinski polygon graphs
For Sierpinski polygon graphs with r not a multiple of 4, isomorphism is classified by a dihedral group action on the defining sequence, and eventually-constant sequences yield exactly two Busemann and countably many non-Busemann horofunctions.