Geodesic
The Groups of Isometries of the Homogeneous Tree and Non-Unimodularity
Bollettino della Unione matematica italiana, Série 9, Tome 6 (2013) no. 3, pp. 565-577.

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In this paper we describe the groups of isometries acting transitively on the homogeneous tree of degree three. This description implies that the following three properties are equivalent: amenability, non-unimodularity and action without inversions. Moreover, we exhibit examples of non-unimodular transitive groups of isometries of a homogeneous tree of degree $q + 1 > 3$ which do not fix any point of the boundary of the tree. 
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Nebbia, Claudio. The Groups of Isometries of the Homogeneous Tree and Non-Unimodularity. Bollettino della Unione matematica italiana, Série 9, Tome 6 (2013) no. 3, pp. 565-577. http://geodesic.mathdoc.fr/item/BUMI_2013_9_6_3_a4/

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