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

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 \(\tilde A_ 2\).

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 |

### Citations:

Zbl 0694.51001
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\textit{D. I. Cartwright} et al., Geom. Dedicata 47, No. 2, 143--166 (1993; Zbl 0784.51010)

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