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Let $\Gamma$ be a discrete subgroup of a connected finite dimensional Lie group $G$. Let $K$ be a maximal compact subgroup of $G$ and denote $X=G/K$. It is well-known that $\Gamma$ acts properly on $X$ via multiplication on the left.

Now suppose that a discrete group $\tilde{\Gamma}$ contains $\Gamma$ as a finite index subgroup (I'm not assuming that $\tilde{\Gamma}$ is a subgroup of $G$.)

When can the action of $\Gamma$ on $X$ be extended to a proper action of $\tilde{\Gamma}$ on $X$? Can you give an example where this is not the case?

Edit: Let my emphasize that I only require the action of $\tilde{\Gamma}$ to be proper (i.e. any continuous action with discrete orbits and finite point stablizers). So the action is allowed to have a finite kernel.

Thanks.

Let $\Gamma$ be a discrete subgroup of a connected finite dimensional Lie group $G$. Let $K$ be a maximal compact subgroup of $G$ and denote $X=G/K$. It is well-known that $\Gamma$ acts properly on $X$ via multiplication on the left.

Now suppose that a discrete group $\tilde{\Gamma}$ contains $\Gamma$ as a finite index subgroup (I'm not assuming that $\tilde{\Gamma}$ is a subgroup of $G$.)

When can the action of $\Gamma$ on $X$ be extended to a proper action of $\tilde{\Gamma}$ on $X$? Can you give an example where this is not the case?

Edit: Let my emphasize that I only require the action of $\tilde{\Gamma}$ on $X$ to be proper (i.e. any continuous action with discrete orbits and finite point stablizers). I am not asking whether $\tilde{\Gamma}$ is a subgroup of $G$. The action of $\tilde{\Gamma}$ on $X$ is allowed to have a finite kernel.

Thanks.

Let $\Gamma$ be a discrete subgroup of a connected finite dimensional Lie group $G$. Let $K$ be a maximal compact subgroup of $G$ and denote $X=G/K$. It is well-known that $\Gamma$ acts properly on $X$ via multiplication on the left.

Now suppose that a discrete group $\tilde{\Gamma}$ contains $\Gamma$ as a finite index subgroup (I'm not assuming that $\tilde{\Gamma}$ is a subgroup of $G$.)

When can the action of $\Gamma$ on $X$ be extended to a proper action of $\tilde{\Gamma}$ on $X$? Can you give an example where this is not the case?

Thanks.

Let $\Gamma$ be a discrete subgroup of a connected finite dimensional Lie group $G$. Let $K$ be a maximal compact subgroup of $G$ and denote $X=G/K$. It is well-known that $\Gamma$ acts properly on $X$ via multiplication on the left.

Now suppose that a discrete group $\tilde{\Gamma}$ contains $\Gamma$ as a finite index subgroup (I'm not assuming that $\tilde{\Gamma}$ is a subgroup of $G$.)

When can the action of $\Gamma$ on $X$ be extended to a proper action of $\tilde{\Gamma}$ on $X$? Can you give an example where this is not the case?

Edit: Let my emphasize that I only require the action of $\tilde{\Gamma}$ to be proper (i.e. any continuous action with discrete orbits and finite point stablizers). So the action is allowed to have a finite kernel.

Thanks.

Let $G$ be a discrete subgroup of a connected finite dimensional Lie group $\Gamma$. Let $K$ be a maximal compact subgroup of $\Gamma$ and denote $X=\Gamma/K$. It is well-known that $G$ acts properly on $X$ via multiplication on the left.

Now suppose that a discrete group $\tilde{G}$ contains $G$ as a finite index subgroup (I'm not assuming that $\tilde{G}$ is a subgroup of $\Gamma$.)

When can the action of $G$ on $X$ be extended to a proper action of $\tilde{G}$ on $X$? Can you give an example where this is not the case?

Thanks.

Let $\Gamma$ be a discrete subgroup of a connected finite dimensional Lie group $G$. Let $K$ be a maximal compact subgroup of $G$ and denote $X=G/K$. It is well-known that $\Gamma$ acts properly on $X$ via multiplication on the left.

Now suppose that a discrete group $\tilde{\Gamma}$ contains $\Gamma$ as a finite index subgroup (I'm not assuming that $\tilde{\Gamma}$ is a subgroup of $G$.)

When can the action of $\Gamma$ on $X$ be extended to a proper action of $\tilde{\Gamma}$ on $X$? Can you give an example where this is not the case?

Thanks.