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When $\Gamma$ is nonelementary, I think the condition is simply that the map is the identity on $\mathbb{R}$. These $\mu$ are the Teichmuller trivial differentials for $\Gamma$.

This should be in Gardiner's "Teichmuller Theory and Quadratic Differentials" somewhere in the chapter on the Teichmuller space of a fuchsian group.

When $\Gamma$ is nonelementary, I think the condition is simply that the map is the identity on the limit set of $\Gamma$.

This should be in Gardiner's "Teichmuller Theory and Quadratic Differentials" somewhere in the chapter on the Teichmuller space of a fuchsian group.

When $\Gamma$ is nonelementary, I think the condition is simply that the map is the identity on the limit set of $\Gamma$. These $\mu$ are the Teichmuller trivial differentials for $\Gamma$.

This should be in Gardiner's "Teichmuller Theory and Quadratic Differentials" somewhere in the chapter on the Teichmuller space of a fuchsian group.

When $\Gamma$ is nonelementary, I think the condition is simply that the map is the identity on $\mathbb{R}$. These $\mu$ are the Teichmuller trivial differentials for $\Gamma$.

This should be in Gardiner's "Teichmuller Theory and Quadratic Differentials" somewhere in the chapter on the Teichmuller space of a fuchsian group.

When $\Gamma$ is nonelementary, I think the condition is simply that the map is the identity on the limit set of $\Gamma$. In particular, when the limit set is all of $\widehat{\mathbb{R}}$, these $\mu$ are the Teichmuller trivial differentials.

This should be in Gardiner's "Teichmuller Theory and Quadratic Differentials" somewhere in the chapter on the Teichmuller space of a fuchsian group.

When $\Gamma$ is nonelementary, I think the condition is simply that the map is the identity on the limit set of $\Gamma$. These $\mu$ are the Teichmuller trivial differentials for $\Gamma$.

This should be in Gardiner's "Teichmuller Theory and Quadratic Differentials" somewhere in the chapter on the Teichmuller space of a fuchsian group.