These groups are countable and discrete, and when n
≥ 3 they are co-compact lattices in PGL(n
+ 1, F) for a local field F. Cartwright and Steger have
constructed an Ãn group in PGL(n + 1, F)
for F =
q((X)),
for any n ≥ 1 and any prime power q.
Only a few examples are known if F =
p.
They have a presentation of a simple type, which makes them easy to work with.
For example, they have Kazhdan's property (T), they are automatic, and the
Cartwright/Steger examples are generated by an automaton. Calculations with some
examples of Ã2 groups led to a suitable definition of
Ramanujan complex of type Ãn. They are associated with
buildings of type Ãn. If F is a local field, there is a
building of type Ãn on the vertices of which PGL(n
+ 1, F) acts transitively. An Ãn group in PGL(n
+ 1, F) acts simply transitively on these vertices.
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