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For a Riemann surface of infinite type, the Teichmiiller modular group does not act properly discontinuously on the Teichmuller space, in general. As an analogy to the theory of Kleinian groups, we divide the Teichmuller space into the limit set and the region of discontinuity for the Teichmiiller modular group, and observe their properties.

Limit sets and regions of discontinuity of Teichmüller modular groups