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On the linearity of lattices in affine buildings and ergodicity of the singular Cartan flow. (English) Zbl 1461.20007

Let \(X\) be a locally finite affine building with a cocompact automorphism group. A lattice \(\Gamma < \operatorname{Aut}(X)\) is called linear if \(\Gamma\) admits a faithful finite-dimensional linear representation over a field, or more generally over a commutative ring with unity.
If \(X\) is the Bruhat-Tits building associated with a semisimple algebraic group \(G\) over a local field \(k\), then the automorphism group of \(X\) is an extension of the group \(G(k)\) by a subgroup of the automorphism group \(\operatorname{Aut}(k)\) of \(k\). The group \(\operatorname{Aut}(k)\) is infinite if and only if the charcteristic of \(k\) is positive, and in this case the authors construct examples of lattices in \(\operatorname{Aut}(X)\) that are not linear. In these examples, the rank of \(X\) equals one, and it is unknown whether examples of higher rank exist.
Since, by a theorem of Tits, every irreducible locally finite affine building of rank at least 3 is Bruhat-Tits, the authors concentrate on the study of buildings of rank 2, in particular those of type \(\tilde{A_2}\). Several examples of lattices in exotic, i.e. not Bruhat-Tits, locally finite buildings of type \(\tilde{A_2}\) are known. The authors establish that such a lattice \(\Gamma\) is necessarily non-linear by showing that any homomorphism \(\Gamma \to \mathrm{GL}_n (R)\) for any commutative unital ring \(R\) and any \(n \ge 1\) has finite image.

MSC:

20E42 Groups with a \(BN\)-pair; buildings
20F65 Geometric group theory
22E40 Discrete subgroups of Lie groups
51E24 Buildings and the geometry of diagrams
22D40 Ergodic theory on groups
20E08 Groups acting on trees
22F50 Groups as automorphisms of other structures
20C99 Representation theory of groups

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