Limit cones of discrete subgroups of reductive groups over a local field (Q1862661)

For a given reductive \(K\)-group \(G= {\mathbf G}(K)\) over a local field \(K\), define \(\Gamma\) as a subgroup of \(G\) and its limit cone as the closed cone generated by the set elements in a Weyl chamber of \(G\) conjugate to the hyperbolic components of \(\Gamma\) [\textit{Y. Benoist}, Geom. Funct. Anal. 7, 1-47 (1997; Zbl 0947.22003)]. If \(K\) is non-Archimedean, it is proved that each closed convex cone with a rational support and stable relative to the opposition involution of a vector Weyl chamber of \(G\) is the limit cone of some Zariski dense subgroup of \(G\).