Skip to main content
MathOverflow

Return to Question

Became Hot Network Question
formatting, changed tags
Source Link
YCor
YCor
  • 63.9k
  • 5
  • 187
  • 286

Centre of orbifold fundamental group of Torustorus (Klein bottle) with one cone point

Let$\newcommand{\orb}{\mathrm{orb}}$Let $T$ ($K$) be the torus (Klein bottle) with one cone point of order $q\geq 2$. The presentation of their orbifold fundamental groups are easy to find. Namely,

$$\pi_1^{orb}(T)=\{a,b\ |\ (aba^{-1}b^{-1})^q=1\}.$$$$\pi_1^{\orb}(T)=\{a,b\ |\ (aba^{-1}b^{-1})^q=1\}.$$

$$\pi_1^{orb}(K)=\{a,b\ |\ (aba^{-1}b)^q=1\}.$$$$\pi_1^{\orb}(K)=\{a,b\ |\ (aba^{-1}b)^q=1\}.$$

Want to know the centre of these two groups. Are they trivial?

Centre of orbifold fundamental group of Torus (Klein bottle) with one cone point

Let $T$ ($K$) be the torus (Klein bottle) with one cone point of order $q\geq 2$. The presentation of their orbifold fundamental groups are easy to find. Namely,

$$\pi_1^{orb}(T)=\{a,b\ |\ (aba^{-1}b^{-1})^q=1\}.$$

$$\pi_1^{orb}(K)=\{a,b\ |\ (aba^{-1}b)^q=1\}.$$

Want to know the centre of these two groups. Are they trivial?

Centre of orbifold fundamental group of torus (Klein bottle) with one cone point

$\newcommand{\orb}{\mathrm{orb}}$Let $T$ ($K$) be the torus (Klein bottle) with one cone point of order $q\geq 2$. The presentation of their orbifold fundamental groups are easy to find. Namely,

$$\pi_1^{\orb}(T)=\{a,b\ |\ (aba^{-1}b^{-1})^q=1\}.$$

$$\pi_1^{\orb}(K)=\{a,b\ |\ (aba^{-1}b)^q=1\}.$$

Want to know the centre of these two groups. Are they trivial?

Source Link
RKS
RKS
  • 585
  • 2
  • 9

Centre of orbifold fundamental group of Torus (Klein bottle) with one cone point

Let $T$ ($K$) be the torus (Klein bottle) with one cone point of order $q\geq 2$. The presentation of their orbifold fundamental groups are easy to find. Namely,

$$\pi_1^{orb}(T)=\{a,b\ |\ (aba^{-1}b^{-1})^q=1\}.$$

$$\pi_1^{orb}(K)=\{a,b\ |\ (aba^{-1}b)^q=1\}.$$

Want to know the centre of these two groups. Are they trivial?