A discrete group is a topological group with the discrete topology. Often in practice, discrete groups arise as discrete subgroups of continuous Lie groups acting on a geometric space. For example, SL_2(Z) is a discrete group, realized as a subgroup of the special linear group SL_2(R) acting on the upper half-plane by Möbius transformations. Discrete groups also appear naturally as symmetries of discrete structures (e.g., graphs, tilings, lattices, polyhedra), fundamental groups of topological spaces, and other related structures.
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