Uniscalar \(p\)-adic Lie groups (Q2716425)

This paper is devoted to the consideration of uniscalar p-adic Lie groups. A locally compact group \(G\) is called uniscalar if each element \(x \in G\) normalizes some open compact subgroup \(U\) of \(G\) (depending on \(x\)). A topological group \(G\) is called pro-discrete if its filter of identity neighbourhoods has a basis of open compact normal subgroups. The main results of this paper are the following: NEWLINENEWLINENEWLINEProposition 3.1. Let \(G\) be a \(p\)-adic Lie group, \(s\) its scale function and \(x \in G\). Then the following conditions are equivalent: (i) \(s(x) = s(x^{-1}) = 1\); (ii) \(x\) normalizes some open compact subgroup; (iii) every identity neighbourhood of \(G\) contains an open compact subgroup normalized by \(x\); (iv) \(\text{Ad}(x) \in \Aut(L(G))\) is a periodic element. NEWLINENEWLINENEWLINECorollary 3.2. Let \(G\) be a \(p\)-adic Lie group. Then \(G\) is uniscalar if and only if \(\text{Ad}(G)\) is a periodic subgroup of \(\Aut(L(G))\). NEWLINENEWLINENEWLINETheorem 5.2. Compactly generated uniscalar \(p\)-adic Lie groups are pro-discrete. The paper contains a counterexample which shows that this theorem is not true if \(G\) is not compactly generated.