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fixed a small typo in the quotient group of RP^2(2,2)
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Neil Hoffman
Neil Hoffman
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There are 10 orbifolds that can be covered by the torus i.e. 10 compact Euclidean 2 orbifolds. However, only 7 of them are quotients of the torus by a cyclic group or Abelian product of cyclic groups. The intuition that Euler characteristics are zero is correct. Formulas for the orbifold Euler characteristic appear throughout the literature. I like Chapter 13 of Thurston's notes in terms of a reference. Specifically, it has a table of all of the 2-orbifolds with non-negative Euler characteristic, which is helpful in this context. Genevieve Walsh's survey Orbifolds and Commensurability is also quite relevant.

The seven orbifolds that are quotients of the torus you are interested in are: $T^2$, the Klein bottle (which can be realized as $T^2/\mathbb{Z}/2\mathbb{Z}$), $S^2(2,2,2,2)$ (which can be also realized as $T^2/\mathbb{Z}/2\mathbb{Z}$ however the group does not act freely in this case), $S^2(2,3,6)$ (which can be realized as $T^2/\mathbb{Z}/6\mathbb{Z}$), $S^2(3,3,3)$ (which can be realized as $T^2/\mathbb{Z}/3\mathbb{Z}$), $S^2(2,4,4)$ (which can be realized as $T^2/\mathbb{Z}/4\mathbb{Z}$), and $RP^2(2,2)$ (which can be realized as $T^2/\mathbb{Z}/(2\mathbb{Z}\times 2\mathbb{Z})$$T^2/(\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z})$). The other 3 compact Euclidean 2 orbifolds are the quotients of the plane by the Euclidean triangle groups. In these cases, the group acting on the torus is dihedral.

There are 10 orbifolds that can be covered by the torus i.e. 10 compact Euclidean 2 orbifolds. However, only 7 of them are quotients of the torus by a cyclic group or Abelian product of cyclic groups. The intuition that Euler characteristics are zero is correct. Formulas for the orbifold Euler characteristic appear throughout the literature. I like Chapter 13 of Thurston's notes in terms of a reference. Specifically, it has a table of all of the 2-orbifolds with non-negative Euler characteristic, which is helpful in this context. Genevieve Walsh's survey Orbifolds and Commensurability is also quite relevant.

The seven orbifolds that are quotients of the torus you are interested in are: $T^2$, the Klein bottle (which can be realized as $T^2/\mathbb{Z}/2\mathbb{Z}$), $S^2(2,2,2,2)$ (which can be also realized as $T^2/\mathbb{Z}/2\mathbb{Z}$ however the group does not act freely in this case), $S^2(2,3,6)$ (which can be realized as $T^2/\mathbb{Z}/6\mathbb{Z}$), $S^2(3,3,3)$ (which can be realized as $T^2/\mathbb{Z}/3\mathbb{Z}$), $S^2(2,4,4)$ (which can be realized as $T^2/\mathbb{Z}/4\mathbb{Z}$), and $RP^2(2,2)$ (which can be realized as $T^2/\mathbb{Z}/(2\mathbb{Z}\times 2\mathbb{Z})$). The other 3 compact Euclidean 2 orbifolds are the quotients of the plane by the Euclidean triangle groups. In these cases, the group acting on the torus is dihedral.

There are 10 orbifolds that can be covered by the torus i.e. 10 compact Euclidean 2 orbifolds. However, only 7 of them are quotients of the torus by a cyclic group or Abelian product of cyclic groups. The intuition that Euler characteristics are zero is correct. Formulas for the orbifold Euler characteristic appear throughout the literature. I like Chapter 13 of Thurston's notes in terms of a reference. Specifically, it has a table of all of the 2-orbifolds with non-negative Euler characteristic, which is helpful in this context. Genevieve Walsh's survey Orbifolds and Commensurability is also quite relevant.

The seven orbifolds that are quotients of the torus you are interested in are: $T^2$, the Klein bottle (which can be realized as $T^2/\mathbb{Z}/2\mathbb{Z}$), $S^2(2,2,2,2)$ (which can be also realized as $T^2/\mathbb{Z}/2\mathbb{Z}$ however the group does not act freely in this case), $S^2(2,3,6)$ (which can be realized as $T^2/\mathbb{Z}/6\mathbb{Z}$), $S^2(3,3,3)$ (which can be realized as $T^2/\mathbb{Z}/3\mathbb{Z}$), $S^2(2,4,4)$ (which can be realized as $T^2/\mathbb{Z}/4\mathbb{Z}$), and $RP^2(2,2)$ (which can be realized as $T^2/(\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z})$). The other 3 compact Euclidean 2 orbifolds are the quotients of the plane by the Euclidean triangle groups. In these cases, the group acting on the torus is dihedral.

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Neil Hoffman
Neil Hoffman
  • 5.3k
  • 1
  • 23
  • 48

There are 10 orbifolds that can be covered by the torus i.e. 10 compact Euclidean 2 orbifolds. However, only 7 of them are quotients of the torus by a cyclic group or Abelian product of cyclic groups. The intuition that Euler characteristics are zero is correct. Formulas for the orbifold Euler characteristic appear throughout the literature. I like Chapter 13 of Thurston's notes in terms of a reference. Specifically, it has a table of all of the 2-orbifolds with non-negative Euler characteristic, which is helpful in this context. Genevieve Walsh's survey Orbifolds and Commensurability is also quite relevant.

The seven orbifolds that are quotients of the torus you are interested in are: $T^2$, the Klein bottle (which can be realized as $T^2/\mathbb{Z}/2\mathbb{Z}$), $S^2(2,2,2,2)$ (which can be also realized as $T^2/\mathbb{Z}/2\mathbb{Z}$ however the group does not act freely in this case), $S^2(2,3,6)$ (which can be realized as $T^2/\mathbb{Z}/6\mathbb{Z}$), $S^2(3,3,3)$ (which can be realized as $T^2/\mathbb{Z}/3\mathbb{Z}$), $S^2(2,4,4)$ (which can be realized as $T^2/\mathbb{Z}/4\mathbb{Z}$), and $RP^2(2,2)$ (which can be realized as $T^2/\mathbb{Z}/(2\mathbb{Z}\times 2\mathbb{Z})$). The other 3 compact Euclidean 2 orbifolds are the quotients of the plane by the Euclidean triangle groups. In these cases, the group acting on the torus is dihedral.