417 research outputs found
Affine actions on non-archimedean trees
We initiate the study of affine actions of groups on -trees for a
general ordered abelian group ; these are actions by dilations rather
than isometries. This gives a common generalisation of isometric action on a
-tree, and affine action on an -tree as studied by I. Liousse. The
duality between based length functions and actions on -trees is
generalised to this setting. We are led to consider a new class of groups:
those that admit a free affine action on a -tree for some .
Examples of such groups are presented, including soluble Baumslag-Solitar
groups and the discrete Heisenberg group.Comment: 27 pages. Section 1.4 expanded, typos corrected from previous versio
Simplicity of some twin tree automorphism groups with trivial commutation relations
We prove simplicity for incomplete rank 2 Kac-Moody groups over algebraic
closures of finite fields with trivial commutation relations between root
groups corresponding to prenilpotent pairs. We don't use the (yet unknown)
simplicity of the corresponding finitely generated groups (i.e., when the
ground field is finite). Nevertheless we use the fact that the latter groups
are just infinite (modulo center).Comment: 10 page
Group-theoretic compactification of Bruhat-Tits buildings
Let GF denote the rational points of a semisimple group G over a
non-archimedean local field F, with Bruhat-Tits building X. This paper contains
five main results. We prove a convergence theorem for sequences of parahoric
subgroups of GF in the Chabauty topology, which enables to compactify the
vertices of X. We obtain a structure theorem showing that the Bruhat-Tits
buildings of the Levi factors all lie in the boundary of the compactification.
Then we obtain an identification theorem with the polyhedral compactification
(previously defined in analogy with the case of symmetric spaces). We finally
prove two parametrization theorems extending the BruhatTits dictionary between
maximal compact subgroups and vertices of X: one is about Zariski connected
amenable subgroups, and the other is about subgroups with distal adjoint
action
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