Can a residually finite group $G \in LFin \rtimes VPoly$ ($G$ is a semi-direct product of a locally finite group by a virtually polycyclic group) have an infinite Hirsch length?
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$\begingroup$ Please provide a definition for Hirsch length (as is usually given only for virtually polycyclic groups). $\endgroup$– YCorCommented Mar 10, 2016 at 22:01
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$\begingroup$ You can use transfinite induction to extend the notion of Hirsch length to elementary amenable groups (see for example secion 1.5 in Four-Manifolds, Geometries and Knots by Hillman). $\endgroup$– Krzysztof SwiecickiCommented Mar 10, 2016 at 22:15
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$\begingroup$ Please give a definition. With transfinite induction I can certainly produce several non-equivalent definitions. $\endgroup$– YCorCommented Mar 10, 2016 at 22:32
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$\begingroup$ For instance, following Definition I.5 in arxiv.org/pdf/1208.1084v3.pdf, it is trivial that (locally-finite)-by-polycyclic-by-finite implies (finite Hirsch length). But there are other possible definitions: a possible definition for finite Hirsch length is that for every ascending or descending chain of subgroups $(H_n)$, eventually $H_n$ has finite index in (or over) $H_{n+1}$. A non-equivalent variant is to assume the same, but only for normal subgroups $H_n$. $\endgroup$– YCorCommented Mar 10, 2016 at 22:49
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$\begingroup$ ... finally Hillman's book is here arxiv.org/pdf/math/0212142v2.pdf... and Theorem 1.8(4) (which just follows the definition) says that $h$ is additive under extensions, and (3) says that $h(N)=0$ for locally finite $N$ (since it's by definition 0 for finite groups). So it is clear that $h$ is finite for (locally finite)-by-(virtually polycyclic) groups. This is not of research level. $\endgroup$– YCorCommented Mar 10, 2016 at 23:30
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