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There are lots of different ways to construct a finitely generated (discrete) group. For example (in no particular order):

1. Fundamental groups of topological spaces;

2. Groups of symmetries of mathematical objects (including fields, manifolds, simplicial, real and $\Lambda$-trees, etc.);

3. Groups given by presentations satisfying various small cancelation conditions (from Tartakovsky to Olshanskii to Gromov);

4. Lattices in Lie groups;

5. Wreath products of various kinds;

6. Direct limits of sequences of groups and their homomorphisms (including various "monsters", etc.);

7. Free constructions (HNN extensions and amalgamated products);

8. Groups simulating various computing devices (there are several different constructions here);

9. Groups acting on locally finite rooted trees (including Grigorchuk groups and iterated monodromy groups of Nekrashevych);

10 Automatic groups,

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There are lots of different ways to construct a finitely generated (discrete) group. For example (in no particular order):

1. Fundamental groups of topological spaces;

2. Groups of symmetries of mathematical objects (including fields, manifolds, simplicial, real and $\Lambda$-trees, etc.);

3. Groups given by presentations satisfying various small cancelation conditions (from Tartakovsky to Olshanskii to Gromov);

4. Lattices in Lie groups;

5. Wreath products of various kinds;

6. Direct limits of sequences of groups and their homomorphisms (including various "monsters", etc.);

7. Free constructions (HNN extensions and amalgamated products);

8. Groups simulating various computing devices (there are several different constructions here);

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