Expansive automorphisms of totally disconnected, locally compact groups

Abstract

Abstract We study topological automorphisms α of a totally disconnected, locally compact group G which are expansive in the sense that ⋂ n ∈ ℤ α n ⁢ ( U ) = { 1 } $\bigcap_{n\in{\mathbb{Z}}}\alpha^{n}(U)=\{1\}$ for some identity neighbourhood U ⊆ G ${U\subseteq G}$ . Notably, we prove that the automorphism induced by an expansive automorphism α on a quotient group G / N ${G/N}$ modulo an α-stable closed normal subgroup N is always expansive. Further results involve the contraction groups U α := { g ∈ G : α n ⁢ ( g ) → 1 as n → ∞ } . $U_{\alpha}:=\{g\in G:\mbox{${\alpha^{n}(g)\to 1}$ as ${n\to\infty}$}\}.$ If α is expansive, then U α ⁢ U α - 1 ${U_{\alpha}U_{\alpha^{-1}}}$ is an open identity neighbourhood in G. We give examples where U α ⁢ U α - 1 ${U_{\alpha}U_{\alpha^{-1}}}$ fails to be a subgroup. However, U α ⁢ U α - 1 ${U_{\alpha}U_{\alpha^{-1}}}$ is an α-stable, nilpotent open subgroup of G if G is a closed subgroup of GL n ⁡ ( ℚ p ) ${\operatorname{GL}_{n}({\mathbb{Q}}_{p})}$ . Further results are devoted to the divisible and torsion parts of U α ${U_{\alpha}}$ , and to the so-called “nub” nub ⁡ ( α ) = U α ¯ ∩ U α - 1 ¯ ${\operatorname{nub}(\alpha)=\overline{U_{\alpha}}\cap\overline{U_{\alpha^{-1}}}}$ of an expansive automorphism.