Fuchsian buildings and BN-pairs (Q5929729)

The buildings in the title of this paper (Fuchsian buildings) are rank 3 hyperbolic buildings whose rank 2 residues are generalized \(n\)-gons for fixed \(n\geq 2\). A certain, rather beautiful, construction of Hagland and Benakli yields buildings \(I(k,L)\) of this type, with \(k\geq 4\) and \(L\) a given generalized \(n\)-gon, \(n\geq 2\). All rank 2 residues of \(I(k,L)\) are isomorphic to \(L\). According to the author, these buildings can also be obtained from a more general construction of Tits, if \(L\) is isomorphic to a finite Pappian polygon, i.e., the finite generalized polygon arising from a simple algebraic group of absolute rank 2 which splits over a finite field. In this case we even get a Moufang building (or half of a twin building, and hence a BN-pair). In the paper under review, the author proves that these buildings admit a BN-pair (acting strongly transitively). Although this might be apparent from \textit{J. Tits}' approach in [J. Alg. 105, 542-573 (1987; Zbl 0626.22013)], the proof in the present paper is very interesting, and it applies to a slightly more general situation, although I could not find an example in the paper that shows that it is really more general (in fact, it is stated as a question in the paper). To achieve the goal, the author provides the reader with some properties of generalized polygons and Moufang polygons. I regret that the author does not mention more classical references for the properties and examples used in the paper. For example, the construction of what is called here an algebraic polygon can best be found in early work of Jacques Tits (and is due to him!), or work of Tits and Bruhat. Also, the proposition 1.3 is a well known fact and proved by the reviewer and \textit{R. Weiss} in [Isr. J. Math. 79, No. 2-3, 321-330 (1992; Zbl 0776.05054)]. Also octagons are neglected here without telling the reason. On page 392, the author asks whether there are non-algebraic polygons with the property that every automorphism (generated by automorphisms fixing rank 1 residues pointwise) fixing a flag can be decomposed into two automorphisms, fixing pointwise the two respective residues determined by the flag. The Moufang octagons are examples, and also some nonclassical quadrangles the smallest of which has order \((3,5)\).