Finding the maximum area of isosceles triangle concept – Følner sequences with eerie shapes Answer

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Let $G$ breathe a discrete and finitely generated group. Recall that ${F_n}_{n in mathbb{N}}$ is a Følner sequence if $|g F_n cup F_n””F_n| rightarrow 1$ for each $g in G$. As is well-known, actuality of a Følner sequence is equal to amenability of $G$.

It is usually mentioned that Følner sequences have peculiar shapes. My tender query is: which examples do we have now that uphold this pretense? Of passage, if $G$ is of subexponential development then a subsequence of balls types a Følner sequence, and this doesn’t have a eerie configuration. Hence, extra particularly: which examples of teams of exponential development do we all know which have categorical Følner sequences not manufactured from balls?

As cases of the examples I’m asking for, Star-shaped Folner sequence asks for Følner units of a inescapable figure, whereas an respond of Folner units and balls provides categorical sequences manufactured from rectangles (versus balls). Likewise, the ax + b group has a Følner sequence manufactured from rectangles the place one facet is exponentially bigger than the opposite.

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