How to prove that the center of the fundamental group of $T_g$ is trivial for $g geq 2$?
10
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Where $T_g$ is a closed orientable surface of genus g.
I want a proof using covering space theory. I know a proof that uses the notion of hyperbolic groups and Riemanian geometry using uniformization theorem. But I want to know is it possible to prove it just using covering space theory?
algebraic-topology manifolds homotopy-theory covering-spaces
asked Mar 3 '17 at 17:58
InfinityInfinity
321112
$endgroup$
add a comment |
10
$begingroup$
Where $T_g$ is a closed orientable surface of genus g.
I want a proof using covering space theory. I know a proof that uses the notion of hyperbolic groups and Riemanian geometry using uniformization theorem. But I want to know is it possible to prove it just using covering space theory?
algebraic-topology manifolds homotopy-theory covering-spaces
asked Mar 3 '17 at 17:58
InfinityInfinity
321112
$endgroup$
$begingroup$
There are purely algebraic proofs of this fact, but you need some background in combinatorial group theory, e.g. in Lyndon-Schupp.
$endgroup$
– Moishe Cohen
Mar 3 '17 at 18:06
$begingroup$
@daw edited....
$endgroup$
– Infinity
Dec 24 '18 at 22:55
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Why do you want that? Fundamental groups themselves are generally nontrivial to compute.
$endgroup$
– anomaly
Dec 24 '18 at 23:35
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@anomaly I don't care about proofs from covering space theory, but a non-hyperbolic proof as in Moishe's first comment would be nice.
$endgroup$
– Mike Miller
Dec 27 '18 at 15:17
add a comment |
10
10
10
3
$begingroup$
Where $T_g$ is a closed orientable surface of genus g.
I want a proof using covering space theory. I know a proof that uses the notion of hyperbolic groups and Riemanian geometry using uniformization theorem. But I want to know is it possible to prove it just using covering space theory?
algebraic-topology manifolds homotopy-theory covering-spaces
asked Mar 3 '17 at 17:58
InfinityInfinity
321112
$endgroup$
Where $T_g$ is a closed orientable surface of genus g.
I want a proof using covering space theory. I know a proof that uses the notion of hyperbolic groups and Riemanian geometry using uniformization theorem. But I want to know is it possible to prove it just using covering space theory?
algebraic-topology manifolds homotopy-theory covering-spaces
algebraic-topology manifolds homotopy-theory covering-spaces
asked Mar 3 '17 at 17:58
InfinityInfinity
321112
asked Mar 3 '17 at 17:58
InfinityInfinity
321112
edited Dec 24 '18 at 22:54
Infinity
asked Mar 3 '17 at 17:58
InfinityInfinity
321112
asked Mar 3 '17 at 17:58
InfinityInfinity
321112
asked Mar 3 '17 at 17:58
InfinityInfinity
321112
321112
$begingroup$
There are purely algebraic proofs of this fact, but you need some background in combinatorial group theory, e.g. in Lyndon-Schupp.
$endgroup$
– Moishe Cohen
Mar 3 '17 at 18:06
$begingroup$
@daw edited....
$endgroup$
– Infinity
Dec 24 '18 at 22:55
$begingroup$
Why do you want that? Fundamental groups themselves are generally nontrivial to compute.
$endgroup$
– anomaly
Dec 24 '18 at 23:35
$begingroup$
@anomaly I don't care about proofs from covering space theory, but a non-hyperbolic proof as in Moishe's first comment would be nice.
$endgroup$
– Mike Miller
Dec 27 '18 at 15:17
add a comment |
$begingroup$
There are purely algebraic proofs of this fact, but you need some background in combinatorial group theory, e.g. in Lyndon-Schupp.
$endgroup$
– Moishe Cohen
Mar 3 '17 at 18:06
$begingroup$
@daw edited....
$endgroup$
– Infinity
Dec 24 '18 at 22:55
$begingroup$
Why do you want that? Fundamental groups themselves are generally nontrivial to compute.
$endgroup$
– anomaly
Dec 24 '18 at 23:35
$begingroup$
@anomaly I don't care about proofs from covering space theory, but a non-hyperbolic proof as in Moishe's first comment would be nice.
$endgroup$
– Mike Miller
Dec 27 '18 at 15:17
$begingroup$
There are purely algebraic proofs of this fact, but you need some background in combinatorial group theory, e.g. in Lyndon-Schupp.
$endgroup$
– Moishe Cohen
Mar 3 '17 at 18:06
$begingroup$
There are purely algebraic proofs of this fact, but you need some background in combinatorial group theory, e.g. in Lyndon-Schupp.
$endgroup$
– Moishe Cohen
Mar 3 '17 at 18:06
$begingroup$
@daw edited....
$endgroup$
– Infinity
Dec 24 '18 at 22:55
$begingroup$
@daw edited....
$endgroup$
– Infinity
Dec 24 '18 at 22:55
$begingroup$
Why do you want that? Fundamental groups themselves are generally nontrivial to compute.
$endgroup$
– anomaly
Dec 24 '18 at 23:35
$begingroup$
Why do you want that? Fundamental groups themselves are generally nontrivial to compute.
$endgroup$
– anomaly
Dec 24 '18 at 23:35
$begingroup$
@anomaly I don't care about proofs from covering space theory, but a non-hyperbolic proof as in Moishe's first comment would be nice.
$endgroup$
– Mike Miller
Dec 27 '18 at 15:17
$begingroup$
@anomaly I don't care about proofs from covering space theory, but a non-hyperbolic proof as in Moishe's first comment would be nice.
$endgroup$
– Mike Miller
Dec 27 '18 at 15:17
add a comment |
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$begingroup$
There are purely algebraic proofs of this fact, but you need some background in combinatorial group theory, e.g. in Lyndon-Schupp.
$endgroup$
– Moishe Cohen
Mar 3 '17 at 18:06
$begingroup$
@daw edited....
$endgroup$
– Infinity
Dec 24 '18 at 22:55
$begingroup$
Why do you want that? Fundamental groups themselves are generally nontrivial to compute.
$endgroup$
– anomaly
Dec 24 '18 at 23:35
$begingroup$
@anomaly I don't care about proofs from covering space theory, but a non-hyperbolic proof as in Moishe's first comment would be nice.
$endgroup$
– Mike Miller
Dec 27 '18 at 15:17