stability results.

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A lattice in a locally compact topological group is a discrete subgroup of finite covolume. Traditionally, lattices arise as arithmetic subgroups in algebraic groups over local fields. There are many strong results in this area, for instance Margulis’ super-rigidity [@Mar:DSS:91] stating roughly that any group homomorphism from a lattice into an algebraic group with non-compact image can be extended to the ambient algebraic group. These strong properties of lattices make their investigation a highly interesting topic.

Algebraic groups over local fields act on affine buildings and it can be shown that the lattice theory of the full automorphism group of the building does not differ much from the lattice theory of the algebraic group.

It is hence natural to extend the theory of lattices in algebraic groups to lattices in the automorphism groups of affine buildings, or even of other polyhedral complexes of non-positive curvature. A survey on results and problems in this direction by Farb, Hruska and Thomas can be found in [@FHT:PAG:09].

Examples of this extension of lattice theory are lattices in trees studied by Bass and Lubotzky in [@BL:TL:01] and lattices in products of trees investigated by Burger and Mozes in [@BM:LPT:00], but also 