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. 
@article{BUMI_2013_9_6_3_a4,
     author = {Nebbia, Claudio},
     title = {The {Groups} of {Isometries} of the {Homogeneous} {Tree} and {Non-Unimodularity}},
     journal = {Bollettino della Unione matematica italiana},
     pages = {565--577},
     year = {2013},
     volume = {Ser. 9, 6},
     number = {3},
     mrnumber = {3202840},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/BUMI_2013_9_6_3_a4/}
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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/