Groups acting simply transitively on the vertices of a building of type \(\tilde A_ 2\). I. (English) Zbl 0784.51010

Cartwright, Donald I.; Mantero, Maria Anna; Steger, Tim; Zappa, Anna

Geom. Dedicata 47, No. 2, 143-166 (1993).

Let \(K\) denote a not necessarily commutative field with discrete valuation. For each \(n \geq 2\) there is a building \(\Delta\) of type \(\tilde A_{n-1}\) on which \({\mathcal G}= \text{PGL}(n,K)\) acts. For the case \(n=2\) see M. Ronan, ‘Lectures on buildings’, Academic Press, New York (1989; Zbl 0694.51001). The authors of the present article consider the case \(n=3\), motivated by the case when \(\Delta\) is the building of \({\mathcal G}= \text{PGL}(3,K)\), \(K\) is a local field, and when \(\Gamma \leq{\mathcal G}\), where \(\Gamma\) is a group acting simply transitively on the vertices of an affine building \(\Delta\).
The authors consider general buildings \(\Delta\) and show that if \(\Delta\) admits a group \(\Gamma\) of automorphisms, then \(\Delta\) must have a diagram of very restricted type. Thereafter, they consider only thick affine buildings \(\Delta\) and assume that their group \(\Gamma\) consists of type-rotating automorphisms of \(\Delta\). They show that there is point-line correspondence \(\lambda:P \to L\) and a triangle presentation \(T\) compatible with \(\lambda\) such that \(\Gamma=\{\{g_ x\}_{x \in P} \mid g_ xg_ yg_ z=1\) for each \((x,y,z) \in T \}\), where \(L\) and \(P\) are sets of neighbours of types 1 and 2, respectively, of a fixed vertex \(v_ 0\) of type 0 and \(g_ x\) is the unique element of \(\Gamma\) such that \(g_ x v_ 0=x\). Then the authors consider the Desarguesian projective plane \((P,L)\) of order \(q\), where \(q\) is any prime power and give a point-line correspondence \(\lambda_ 0:P \to L\) and compatible triangle representation \(T_ 0\). Modifying construction of Tits they embed the associated group \(\Gamma_ 0\) in \({\mathcal G}=\text{PGL} \biggl( 3,F_ q \bigl(( X) \bigr) \biggr)\), where it acts simply transitively on vertices of \(\Delta\). They construct all triangle presentations whose symmetry groups are similarly large as symmetry group of \(T_ 0\) and suppose that it is not clear whether all associate groups \(\Gamma_ T\) arise as subgroups of matrix groups. It is essential that triangle presentations mentioned above give rise to some new building of type \(\tilde A_ 2\).

Reviewer: P.Burda (Ostrava)

MSC:

51E24	Buildings and the geometry of diagrams
51E15	Finite affine and projective planes (geometric aspects)
20G25	Linear algebraic groups over local fields and their integers
20F05	Generators, relations, and presentations of groups

Keywords:

simply transitively acting groups; buildings of type \(\tilde A_ 2\)

Citations:

Zbl 0694.51001

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