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For questions requesting examples of a certain structure or phenomenon
7
votes
books well-motivated with explicit examples
Trees by J-P Serre. The first half is pretty much all theory, but in the second he looks at the explicit example of $SL_2$.
12
votes
Accepted
Torsion-free virtually free-by-cyclic groups
The group $$G=\langle a, b, x, y\mid [a, b]^2=[x, y]^2\rangle$$
is a torsion-free group which is not free by cyclic. However, $G$ is free-by-$D_{\infty}$ and so virtually free-by-cyclic (containing an …