In his PhD thesis the author investigates various compactifications of affine buildings and compares these compactifications. The prime tool is the Busemann compactification Bus\((X,d)\) of a metric space \(X\) with metric \(d\), whose construction and properties are reviewed in the first chapter. It is obtained as the closure in the space of continuous functions on \(X\) of the set of Busemann functions \(b_y (x):z\mapsto d(x,z)-d(x,y)\) with basepoint \(y\in X\). This is the same as the visual compactification where the boundary of \(X\) is identified with geodesic rays from a basepoint \(y\).
The next chapter deals with the concrete situation of finite-dimensional normed real vector spaces \(V\) and their Busemann compactifications. The author considers smooth norms, polyhedral norms (where the unit spheres are polyhedrons) and characteristic norms associated with root systems \(\Phi\) (that is, norms which are invariant under the operation of \(\operatorname{Aut}(\Phi))\). All smooth norms on \(V\) (and in particular the usual \(p\)-norms for \(1< p<\infty\)) lead to isomorphic Busemann compactifications of \(V\). The Busemann compactification with respect to a polyhedral norm is isomorphic to a polyhedral compactification associated with the cone decomposition of \(V\) obtained from the given polyhedral norm. The usual \(p\)-norms for \(1\leq p< \infty\) give rise to characteristic norms \(\| v\| _{\Phi,p}=(\sum_{\alpha\in\Phi} | \langle v,\alpha\rangle| ^2)^{1/p}\) and \(\| v\| _{\Phi,\infty}=\max_{\alpha\in\Phi} | \langle v,\alpha\rangle| \) for \(p=\infty\), which are analysed in detail. The norms for \(1<p<\infty\) are smooth and the norms for \(p=1,\infty\) are polyhedral. In case \(p=1\) the Busemann compactification is isomorphic to E. Landvogt’s polyhedral compactification [A compactification of the Bruhat-Tits building. Lecture Notes in Mathematics, 1619. Berlin: Springer-Verlag (1996; Zbl 0935.20034)]. As an illustration, a root system of type \(B_2\) and the norms \(\| \cdot\| _{\Phi,p}\) for \(p=1,2,\infty\) and \(\| \cdot\| _{\Phi,1}+\| \cdot\| _{\Phi,2}\) are considered.
Chapter 3 is devoted to the introduction of affine buildings. In order to allow for buildings obtained from semisimple algebraic groups over local fields, the author uses polysimplicial complexes instead of simplicial ones. It essentially follows the ideas developed by F. Bruhat and J. Tits [Publ. Math., Inst. Hautes Étud. Sci. 41, 5–251 (1972; Zbl 0254.14017)] but care is taken to give an intrinsic definition and to distinguish between a complex and its realization in a topological space.
The following chapter deals with the Busemann compactifications of affine buildings \(X\) with respect to characteristic metrics \(d\) on \(X\), that is, for each apartment \(\Sigma\) of \(X\) the restriction of \(d\) onto \(\Sigma\) is induced by a characteristic norm associated with \(\Phi\) on an associated vector space \(V\) (such that \(\Sigma\cong\Sigma(W,V)\) where \(W\) is the affine Weyl group of \(\Phi\)). For each of the characteristic norms \(\| \cdot\| _{\Phi,p}\) for \(1\leq p\leq \infty\) one obtains a characteristic metric \(d_p\) on \(X\), \(d_2\) being up to a scalar factor the metric normally used on affine buildings. The author shows that Bus\((X,d)\) is the union of all closures of apartments containing a fixed chamber \(C\) and that for any two points of Bus\((X,d)\) there is an apartment whose closure contains both points. The author further gathers properties of Busemann compactifications of locally finite affine buildings that admit a topological group of continuous automorphisms acting transitively on the set of incident chamber-apartment pairs. (Such buildings, for example, arise from linear algebraic groups over local fields.)
In the last chapter the author investigates the Busemann compactifications of affine buildings \(X\) with respect to the characteristic metrics \(d_p\) for \(1\leq p\leq\infty\). In these cases the entire boundary of \(X\) is visible from any point of \(X\). If \(X\) is the affine building to a semisimple linear algebraic group over a non-archimedean local field then Bus\((X,d_1)\) is isomorphic to the Landvogt compactification and the boundary of Bus\((X,d_2)\) isomorphic to the Borel-Serre compactification of \(X\).
Alhough many known compactifications of affine buildings are isomorphic to certain Busemann compactifications the author provides an example where this is not the case. He shows that for \(n\geq 2\) A. Werner’s compactification of the building \(X\) corresponding to PGL\(_n(K)\) over a non-archimedean local field \(K\) from [Math. Z. 248, No. 3, 511–526 (2004; Zbl 1121.20024)] is not isomorphic to any Busemann compactification of \(X\) with respect to a characteristic metric but that it is dominated by Bus\((X,d_1)\).