Embeddings of twin trees (Q1290881)

A generalized \(n\)-gon is a building of rank 2 with finite diameter. The buildings of rank 2 with infinite diameter are the trees. But almost nothing of the beautiful theory of generalized polygons extends to trees. Alternatively, generalized polygons are also twin buildings of rank 2. In this point of view, their analogues of infinite diameter are the twin trees. Thanks to the opposition relation, these resemble more closely their relatives of finite diameter. For instance, with every thick generalized polygon there is associated a pair of cardinal numbers, the order \((s,t)\) of the polygon. It just means that there are a constant number of points incident with a line (and the constant is \(s+1)\), and there are a constant number of lines incident with any point (namely, \(t+1)\). This is also true for twin trees, where the pair \((s+1,t+1)\) is the bidegree of the two trees. In the finite case, however, there are strong restrictions on the numbers \(s,t,s',t'\) if a thick generalized \(n\)-gon with order \((s',t')\) is embedded in a thick generalized \(n\)-gon with order \((s,t)\). These restrictions depend on \(n\), but they generally say that \(st\) must be much larger than \(s't'\). In the paper under review, the authors show that this has no analogue for twin trees. Indeed, they prove that every twin tree with bidegree \((s'+1,t'+1)\) is embedded in a twin tree of bidegree \((s+1,t+1)\) whenever \(s\geq s'\) and \(t\geq t'\). To achieve this, they prove a generalization of a result of \textit{M. A. Ronan} and \textit{J. Tits} [see Proposition 6 of Isr. J. Math. 109, 349-377 (1999; Zbl 0924.20024)].