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That is not an answer. I want to give an example where the argument of Erdős does not work directly.

Consider an action of group $\Gamma$ on $\mathbb R^3$ generated by the reflections $r_1, r_2$ and $r_3$ correspondingly in the lines $x=z=0$ and $x+1=z=0$ and $x-y=z-1=0$.

Each of the reflections $r_i$ generate a maxiamal $\mathbb Z_2$-subgroups, all of them are nonconjugate. These groups corespond to three singular circles, say $\Sigma_i$ in the factor $X=\mathbb R^3/\Gamma$. ($X$ is homeomorphic to $S^3$ and $\Sigma_1$, $\Sigma_2$, $\Sigma_3$ form Borromean rings, but all this is not important.)

Let us try to mimic argument of Erdős. Take subsets $X_i$ of $X$ of midpoints $m$ between $x\in X$ and a closest $x_0\in\Sigma_i$ to $x$. As in the argument of Erdős we have* $\mathrm{vol}\\, X_i>\tfrac{1}{2^3}\cdot\mathrm{vol}\\, X$. BUT $X_1\cap X_3$ has interior points and here argument brakes into parts.

Comments

• Since fixed point sets are 1-dimensional, it would be enough to take $m\in [xx_0]$ such that $\tfrac{|mx_0|}{|xx_0|}=\tfrac1{2\sqrt[3]{2}}$. But even in this case one has interior points in $X_1\cap X_2$ (the borderline in this example seems to be $\tfrac13$).

• There is a natural bisecting hyperplane for any two affine subspaces. We may use it to cut a cylinder domain around each fixed point set of a maximal subgroup. The projection of these cylinders in $X$ gives Voronoi-like domains, but they do not cover whole space in general --- that is OK as far as we have lower bound on their volumes...

That is not an answer. I want to give an example where the argument of Erdős does not work directly.

Consider an action of group $\Gamma$ on $\mathbb R^3$ generated by the reflections $r_1, r_2$ and $r_3$ correspondingly in the lines $x=z=0$ and $x+1=z=0$ and $x-y=z-1=0$.

Each of the reflections $r_i$ generate a maxiamal $\mathbb Z_2$-subgroups, all of them are nonconjugate. These groups corespond to three singular circles, say $\Sigma_i$ in the factor $X=\mathbb R^3/\Gamma$. ($X$ is homeomorphic to $S^3$ and $\Sigma_1$, $\Sigma_2$, $\Sigma_3$ form Borromean rings, but all this is not important.)

Let us try to mimic argument of Erdős. Take subsets $X_i$ of $X$ of midpoints $m$ between $x\in X$ and a closest $x_0\in\Sigma_i$ to $x$. As in the argument of Erdős we have $\mathrm{vol}\, X_i>\tfrac{1}{2^3}\cdot\mathrm{vol}\, X$. BUT $X_1\cap X_3$ has interior points and here argument brakes into parts.

Comments

• Since fixed point sets are 1-dimensional, it would be enough to take $m\in [xx_0]$ such that $\tfrac{|mx_0|}{|xx_0|}=\tfrac1{2\sqrt[3]{2}}$. But even in this case one has interior points in $X_1\cap X_2$ (the borderline in this example seems to be $\tfrac13$).

• There is a natural bisecting hyperplane for any two affine subspaces. We may use it to cut a cylinder domain around each fixed point set of a maximal subgroup. The projection of these cylinders in $X$ gives Voronoi-like domains, but they do not cover whole space in general --- that is OK as far as we have lower bound on their volumes...

That is not an answer. I want to give an example where the argument of Erdős does not work directly.

Consider an action of group $\Gamma$ on $\mathbb R^3$ generated by the reflections $r_1, r_2$ and $r_3$ correspondingly in the lines $x=z=0$ and $x+1=z=0$ and $x-y=z-1=0$.

Each of the reflections $r_i$ generate a maxiamal $\mathbb Z_2$-subgroups, all of them are nonconjugate. These groups corespond to three singular circles, say $\Sigma_i$ in the factor $X=\mathbb R^3/\Gamma$.

Let us try to mimic argument of Erdős. Take subsets $X_i$ of $X$ of midpoints $m$ between $x\in X$ and a closest $x_0\in\Sigma_i$ to $x$. As in the argument of Erdős we have* $\mathrm{vol}\\, X_i>\tfrac{1}{2^3}\cdot\mathrm{vol}\\, X$. BUT $X_1\cap X_3$ has interior points and here argument brakes into parts.

Comments

• Since fixed point sets are 1-dimensional, it would be enough to take $m\in [xx_0]$ such that $\tfrac{|mx_0|}{|xx_0|}=\tfrac1{2\sqrt[3]{2}}$. But even in this case one has interior points in $X_1\cap X_2$ (the borderline in this example seems to be $\tfrac13$).

• There is a natural bisecting hyperplane for any two affine subspaces. We may use it to cut a cylinder domain around each fixed point set of a maximal subgroup. The projection of these cylinders in $X$ gives Voronoi-like domains, but they do not cover whole space in general --- that is OK as far as we have lower bound on their volumes...

That is not an answer. I want to give an example where the argument of Erdős does not work directly.

Consider an action of group $\Gamma$ on $\mathbb R^3$ generated by the reflections $r_1, r_2$ and $r_3$ correspondingly in the lines $x=z=0$ and $x+1=z=0$ and $x-y=z-1=0$.

Each of the reflections $r_i$ generate a maxiamal $\mathbb Z_2$-subgroups, all of them are nonconjugate. These groups corespond to three singular circles, say $\Sigma_i$ in the factor $X=\mathbb R^3/\Gamma$. ($X$ is homeomorphic to $S^3$ and $\Sigma_1$, $\Sigma_2$, $\Sigma_3$ form Borromean rings, but all this is not important.)

Let us try to mimic argument of Erdős. Take subsets $X_i$ of $X$ of midpoints $m$ between $x\in X$ and a closest $x_0\in\Sigma_i$ to $x$. As in the argument of Erdős we have* $\mathrm{vol}\\, X_i>\tfrac{1}{2^3}\cdot\mathrm{vol}\\, X$. BUT $X_1\cap X_3$ has interior points and here argument brakes into parts.

Comments

• Since fixed point sets are 1-dimensional, it would be enough to take $m\in [xx_0]$ such that $\tfrac{|mx_0|}{|xx_0|}=\tfrac1{2\sqrt[3]{2}}$. But even in this case one has interior points in $X_1\cap X_2$ (the borderline in this example seems to be $\tfrac13$).

• There is a natural bisecting hyperplane for any two affine subspaces. We may use it to cut a cylinder domain around each fixed point set of a maximal subgroup. The projection of these cylinders in $X$ gives Voronoi-like domains, but they do not cover whole space in general --- that is OK as far as we have lower bound on their volumes...

That is not an answer. I want to give an example where the argument of Erdős does not work directly.

Consider an action of group $\Gamma$ on $\mathbb R^3$ generated by the reflections $r_1, r_2 and $r_3$ correspondingly in the lines $x=z=0$ and $x+1=z=0$ and $x-y=z-1=0$.

Each of the reflections $r_i$ generate a maxiamal $\mathbb Z_2$-subgroups, all of them are nonconjugate. These groups corespond to three singular circles, say $\Sigma_i$ in the factor $X=\mathbb R^3/\Gamma$.

Let us try to mimic argument of Erdős. Take subsets $X_i$ of $X$ of midpoints $m$ between $x\in X$ and a closest $x_0\in\Sigma_i$ to $x$. As in the argument of Erdős we have* $\mathrm{vol}\\, X_i>\tfrac{1}{2^3}\cdot\mathrm{vol}\\, X$. BUT $X_1\cap X_3$ has interior points and here argument brakes into parts.

Comments

• Since fixed point sets are 1-dimensional, it would be enough to take $m\in [xx_0]$ such that $\tfrac{|mx_0|}{|xx_0|}=\tfrac1{2\sqrt[3]{2}}$. But even in this case one has interior points in $X_1\cap X_2$ (the borderline in this example seems to be $\tfrac13$).

• There is a natural bisecting hyperplane for any two affine subspaces. We may use it to cut a cylinder domain around each fixed point set of a maximal subgroup. The projection of these cylinders in $X$ gives Voronoi-like domains, but they do not cover whole space in general --- that is OK as far as we have lower bound on their volumes...

That is not an answer. I want to give an example where the argument of Erdős does not work directly.

Consider an action of group $\Gamma$ on $\mathbb R^3$ generated by the reflections $r_1, r_2$ and $r_3$ correspondingly in the lines $x=z=0$ and $x+1=z=0$ and $x-y=z-1=0$.

Each of the reflections $r_i$ generate a maxiamal $\mathbb Z_2$-subgroups, all of them are nonconjugate. These groups corespond to three singular circles, say $\Sigma_i$ in the factor $X=\mathbb R^3/\Gamma$.

Let us try to mimic argument of Erdős. Take subsets $X_i$ of $X$ of midpoints $m$ between $x\in X$ and a closest $x_0\in\Sigma_i$ to $x$. As in the argument of Erdős we have* $\mathrm{vol}\\, X_i>\tfrac{1}{2^3}\cdot\mathrm{vol}\\, X$. BUT $X_1\cap X_3$ has interior points and here argument brakes into parts.

Comments

• Since fixed point sets are 1-dimensional, it would be enough to take $m\in [xx_0]$ such that $\tfrac{|mx_0|}{|xx_0|}=\tfrac1{2\sqrt[3]{2}}$. But even in this case one has interior points in $X_1\cap X_2$ (the borderline in this example seems to be $\tfrac13$).

• There is a natural bisecting hyperplane for any two affine subspaces. We may use it to cut a cylinder domain around each fixed point set of a maximal subgroup. The projection of these cylinders in $X$ gives Voronoi-like domains, but they do not cover whole space in general --- that is OK as far as we have lower bound on their volumes...