A class of well-founded totally disconnected locally compact groups

Abstract

Motivated by the problem of finding a "well-foundedness principle" for totally disconnected, locally compact (t.d.l.c.) groups, we introduce a class $\mathscr{E}^{\mathscr{S}}$ of t.d.l.c. groups, containing P. Wesolek's class $\mathscr{E}$ of (regionally) elementary groups but also including many groups in the class $\mathscr{S}$ of nondiscrete compactly generated topologically simple t.d.l.c. groups. The class $\mathscr{E}^{\mathscr{S}}$ carries a well-behaved rank function and is closed under taking directed unions, open subgroups, closed normal subgroups, extensions and quotients. The class $\mathscr{E}^{\mathscr{S}}$ also includes other well-studied families of t.d.l.c. groups that are not contained in $\mathscr{E}$, including all locally linear t.d.l.c. groups, all complete geometric Kac--Moody groups over finite fields, the Burger--Mozes groups $U(F)$ where $F$ is primitive, and $2^{\aleph_0}$ more examples of groups in $\mathscr{S}$ that arise as groups acting on trees with Tits' independence property (P). On the other hand, $\mathscr{E}^{\mathscr{S}}$ excludes the Burger--Mozes groups $U(F)$ where $F$ is nilpotent and does not act freely. By contrast, a larger class $\mathscr{E}^{[\mathrm{Sim}]}$ (with similar closure properties to $\mathscr{E}^{\mathscr{S}}$) is closed under forming actions on trees with property (P).