In 2000, M. Burger and S. Mozes introduced universal groups acting on trees with a prescribed local action. We generalize this concept to groups acting on right-angled buildings. When the right-angled building is thick and irreducible of rank at least 2 and each of the local permutation groups is transitive and generated by its point stabilizers, we show that the corresponding universal group is a simple group.When the building is locally finite, these universal groups are compactly generated totally disconnected locally compact groups, and we describe the structure of the maximal compact open subgroups of the universal groups as a limit of generalized wreath products.
We provide a uniform framework to study the exceptional homogeneous compact geometries of type C3. This framework is then used to show that these are simply connected, answering a question by Kramer and Lytchak, and to calculate the full automorphism groups.MSC 2010: 51E24, 57S15
In this paper we prove equivalence of sets of axioms for nondiscrete affine buildings, by providing different types of metric, exchange and atlas conditions. We apply our result to show that the definition of a Euclidean building depends only on the topological equivalence class of the metric on the model space. The sharpness of the axioms dealing with metric conditions is illustrated in an appendix. There it is shown that a space X defined over a model space with metric d is possibly a building only if the induced distance function on X satisfies the triangle inequality.
In this paper, we show that the building at infinity of a two-dimensional affine R-building is a generalized polygon endowed with a valuation satisfying some specific axioms. Specializing to the discrete case of affine buildings, this solves part of a long standing conjecture about affine buildings of type e G2, and it reproves the results obtained mainly by the second author for types e A2 and e C2. The techniques are completely different from the ones employed in the discrete case, but they are considerably shorter, and general (i.e., independent of the type of the two-dimensional R-building).
Abstract. We prove two generalizations of results of Bruhat and Tits involving metrical completeness and R-buildings. Firstly, we give a generalization of the Bruhat-Tits fixed point theorem also valid for non-complete R-buildings with the added condition that the group is finitely generated.Secondly, we generalize a criterion which reduces the problem of completeness to the wall trees of the R-building. This criterion was proved by Bruhat and Tits for R-buildings arising from root group data with valuation.
Mathematics Subject Classification (2010). 51E24, 20F65.
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