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Palindromic subshifts and simple periodic groups of intermediate growth. (English) Zbl 1437.20038

This is a groundbreaking article, solving fundamental questions from group theory by introducing “soft” substitutes of algebraic arguments.
The main result is that there exist infinite, finitely generated simple groups that are torsion and of intermediate word growth.
The author, more precisely, examines the following situation. Consider a minimal action of the infinite dihedral group \(\langle a,b\rangle\) on the Cantor set \(\mathcal X\). A fragmentation of the generator \(a\) is an elementary \(2\)-group \(A\) of homeomorphisms of \(\mathcal X\) such that for every \(\zeta\in\mathcal X\) we have \(A(\zeta)=\{\zeta,a(\zeta)\}\); let similarly \(B\) be a fragmentation of \(B\), and set \(G=\langle A,B\rangle\).
Theorem 1.1. Suppose that \(\xi\) is a fixed point of \(a\) and that for every \(h\in A\) fixing \(\xi\), the interior of the set of fixed points of \(h\) accumulates on \(\xi\). Then the group \(G\) is periodic (aka torsion) and infinite.
Note that every orbit of \(\langle a,b\rangle\) on \(\mathcal X\) is a line or half-line, properly coloured by alternating \(a\)’s and \(b\)’s; and the orbits of \(G\) are likewise coloured by alternating letters in \(A\) and \(B\). This colouring is linearly repetitive if there is a constant \(C\) such that every \(A,B\)-coloured segment of width \(\ell\) that appears in a line appears in every ball of radius \(C \ell\) in every line.
Theorem 1.2. Under the assumptions of Theorem 1.1, if the colouring of orbits is linearly repetitive then the group \(G\) has intermediate word growth (namely, the number of group elements expressible as a product of at most \(n\) generators grows both superpolynomially and subexponentially in \(n\)).
Recall that the topological full group of a group \(G\) acting on \(\mathcal X\) consists in all homeomorphisms of \(\mathcal X\) that agree locally with elements of \(G\). The subgroup \(\mathsf A(G)\) of the topological full group is generated by all homeomorphisms mapping \(U_1\to U_2\to U_3\to U_1\) via elements of \(G\) and fixing the complement. If the action of \(G\) on \(\mathcal X\) is expansive, then \(\mathsf A(G)\) is finitely generated and simple.
Theorem 1.2, continued. Under the conditions of Theorem 1.2, the group \(\mathsf A(G)\) is a finitely generated, simple torsion group of intermediate word growth.
The main technique of proof consists in controlling the size of “inverted orbits”, namely, for a word \(w=s_1\cdots s_M\) with \(s_i\in A\cup B\) and a point \(\zeta\in\mathcal X\), in showing that the size of \(O_{w,\zeta}:=\{s_1\cdots s_i\zeta:1\le i\le M\}\) grows sublinearly. This is done by clever arguments crucially relying on the linear shape of the orbits of \(G\): either \(O_w\) is small, or the forward path \((s_i\cdots s_M\zeta)\) has to pass through many “gateways” at which it turns round for many other \(\zeta'\). These arguments are very flexible substitutes for the more classical induction arguments based on length-shortening and self-similarity.
The method covers in particular the previous constructions by Grigorchuk [Gri80, Gri83 {from the text’s references}] and others. In his last section, the author describes quite explicitly an example based on the action of \(D_\infty\) on the “Fibonacci shift”, which is a concrete example of finitely generated, simple torsion group of intermediate word growth.

MSC:

20F69 Asymptotic properties of groups
20F50 Periodic groups; locally finite groups
20E08 Groups acting on trees
20E32 Simple groups
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