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Let G be a regular Lie group which is a directed union of regular Lie groups Gi (all modelled on possibly infinite‐dimensional, locally convex spaces). We show that G=lim⟶Gi as a regular Lie group if G admits a so‐called direct limit chart. Notably, this allows the regular Lie group Diffc(M) of compactly supported diffeomorphisms to be interpreted as a direct limit of the regular Lie groups DiffK(M) of diffeomorphisms supported in compact sets K⊆M, even if the finite‐dimensional smooth manifold M is merely paracompact (but not necessarily σ‐compact), which is new. Similar results are obtained for the test function groups Cck(M,F) with values in a Lie group F. [ABSTRACT FROM AUTHOR]

The class of elementary totally disconnected groups is the smallest class of totally disconnected, locally compact, second countable groups which contains all discrete countable groups, all metrizable pro-finite groups, and is closed under extensions and countable ascending unions. To each elementary group G, a (possibly infinite) ordinal number rk(G) can be associated, its construction rank. By a structure theorem of Phillip Wesolek, elementary p-adic Lie groups are among the basic building blocks for general s-compact p-adic Lie groups. We characterize elementary p-adic Lie groups in terms of the subquotients needed to describe them. The characterization implies that every elementary p-adic Lie group has finite construction rank. Structure theorems concerning general p-adic Lie groups are also obtained. [ABSTRACT FROM AUTHOR]

We study topological automorphisms α of a totally disconnected, locally compact group G which are expansive in the sense that ... for some identity neighbourhood U⊆G. Notably, we prove that the automorphism induced by an expansive automorphism α on a quotient group G/N modulo an α-stable closed normal subgroup N is always expansive. Further results involve the contraction groups Uα:={g∞G:αn(g)→1asn→∞}. If α is expansive, then UαUα-1 is an open identity neighbourhood in G. We give examples where UαUα-1 fails to be a subgroup. However, UαUα-1 is an α-stable, nilpotent open subgroup of G if G is a closed subgroup of GLn(...). Further results are devoted to the divisible and torsion parts of Uα, and to the so-called "nub" nub(α)=... of an expansive automorphism. [ABSTRACT FROM AUTHOR]