Why should tropical moduli spaces be stacks? A topological viewpoint.

From CRCG-Wiki

The classical moduli space of complex elliptic curves, as a topological space, is boring: it's just the complex plane. However, elliptic curves have automorphisms, and so we should consider it not as a space but as an orbifold, or stack. If we do this, the topology of the moduli space of elliptic curves becomes interesting: it has fundamental group SL_2(Z), and its universal cover is the upper half plane, which is contractible. So studying the moduli stack of elliptic curves is essentially equivalent to studying the group SL_2(Z).

This beautiful example is the starting point of huge amounts of on three separate families of groups: arithmetic groups such as GL_d(Z), mapping class groups of genus g surfaces Mod(S_g), and the groups Aut(F_n) and Out(F_n) of automorphisms and outer automorphisms of free groups. I aim to give a gentle overview of the story of studying these families of groups in order to give a topological reason of why we should consider tropical moduli spaces as stacks, following a suggestion of Mikhalkin. In short: if we do, then the fundamental group of the moduli space of tropical curves of genus g is Out(F_g), and the fundamental group of the moduli space of tropical abelian varieties of dimension d is GL_d(Z).