Mixing
Word length vs hyperbolic length of curves on a hyperbolic surface
$begingroup$
Suppose S is a surface which admits a hyperbolic metric - by this I mean a complete Riemannian metric of constant negative curvature, with totally geodesic (possibly empty) boundary.
Fix some finite presentation of the fundamental group $$pi_1(S) = langle X | R rangle.$$
For an element $gin pi_1(S)$, define the word length $|w|_X$, with respect to the generating set X, to be the minimum number of generators needed to express $w$ as a word in X.
Fact: Every homotopy class of curves in $S$ contains a unique geodesic.
Question: What's the relationship between the word length of $w$, and the hyperbolic length of the corresponding geodesic?
riemann-surfaces hyperbolic-geometry geometric-group-theory low-dimensional-topology
asked Jan 10 at 19:13
38917803891780
816
$endgroup$
|
show 4 more comments
$begingroup$
Suppose S is a surface which admits a hyperbolic metric - by this I mean a complete Riemannian metric of constant negative curvature, with totally geodesic (possibly empty) boundary.
Fix some finite presentation of the fundamental group $$pi_1(S) = langle X | R rangle.$$
For an element $gin pi_1(S)$, define the word length $|w|_X$, with respect to the generating set X, to be the minimum number of generators needed to express $w$ as a word in X.
Fact: Every homotopy class of curves in $S$ contains a unique geodesic.
Question: What's the relationship between the word length of $w$, and the hyperbolic length of the corresponding geodesic?
riemann-surfaces hyperbolic-geometry geometric-group-theory low-dimensional-topology
asked Jan 10 at 19:13
38917803891780
816
$endgroup$
1
$begingroup$
In group terms, define in any finitely generated group with given word metric, $c(g)$ as $min_h|hgh^{-1}|$. Then in your setting the length of the geodesic defined by $w$ is comparable to $c(w)$. (All this is unrelated to the choice of relators. Only the choice of generators matters.)
$endgroup$
– YCor
Jan 10 at 19:43
$begingroup$
I'm not sure I understand the question. There are lots of hyperbolic metrics on your surface, that don't change the group, and lots of presentations, that don't change the metric. What kind of relationship are you looking for?
$endgroup$
– Hempelicious
Jan 10 at 23:09
1
$begingroup$
You are confusing/conflating based homotopy classes (aka elements of $G=pi_1(S,x)$) and free homotopy classes (aka conjugacy classes in $G$). Which do you really mean? If you mean based homotopy classes then the relation of the word length and the hyperbolic length is given by Milnor-Schwarz lemma.
$endgroup$
– Moishe Cohen
Jan 11 at 0:57
$begingroup$
@MoisheCohen isn't it true that free homotopy classes have a one-to-one correspondence with conjugacy classes of $pi_1(S,x)$? And could you expand on the significance of the Milnor-Schwartz lemma in this case?
$endgroup$
– 3891780
Jan 11 at 10:36
1
$begingroup$
@3891780: 1. Yes, this is what I said in my comment. 2. I could, but it is unclear to me what your question really is. Are you talking about based or unbased (free) homotopy classes? Do you know what Milnor-Schwarz lemma is? Do you know about Morse lemma (stability of geodesics)?
$endgroup$
– Moishe Cohen
Jan 11 at 18:06
|
show 4 more comments
$begingroup$
Suppose S is a surface which admits a hyperbolic metric - by this I mean a complete Riemannian metric of constant negative curvature, with totally geodesic (possibly empty) boundary.
Fix some finite presentation of the fundamental group $$pi_1(S) = langle X | R rangle.$$
For an element $gin pi_1(S)$, define the word length $|w|_X$, with respect to the generating set X, to be the minimum number of generators needed to express $w$ as a word in X.
Fact: Every homotopy class of curves in $S$ contains a unique geodesic.
Question: What's the relationship between the word length of $w$, and the hyperbolic length of the corresponding geodesic?
riemann-surfaces hyperbolic-geometry geometric-group-theory low-dimensional-topology
asked Jan 10 at 19:13
38917803891780
816
$endgroup$
Suppose S is a surface which admits a hyperbolic metric - by this I mean a complete Riemannian metric of constant negative curvature, with totally geodesic (possibly empty) boundary.
Fix some finite presentation of the fundamental group $$pi_1(S) = langle X | R rangle.$$
For an element $gin pi_1(S)$, define the word length $|w|_X$, with respect to the generating set X, to be the minimum number of generators needed to express $w$ as a word in X.
Fact: Every homotopy class of curves in $S$ contains a unique geodesic.
Question: What's the relationship between the word length of $w$, and the hyperbolic length of the corresponding geodesic?
riemann-surfaces hyperbolic-geometry geometric-group-theory low-dimensional-topology
riemann-surfaces hyperbolic-geometry geometric-group-theory low-dimensional-topology
asked Jan 10 at 19:13
38917803891780
816
asked Jan 10 at 19:13
38917803891780
816
asked Jan 10 at 19:13
38917803891780
816
asked Jan 10 at 19:13
38917803891780
816
asked Jan 10 at 19:13
38917803891780
816
816
1
$begingroup$
In group terms, define in any finitely generated group with given word metric, $c(g)$ as $min_h|hgh^{-1}|$. Then in your setting the length of the geodesic defined by $w$ is comparable to $c(w)$. (All this is unrelated to the choice of relators. Only the choice of generators matters.)
$endgroup$
– YCor
Jan 10 at 19:43
$begingroup$
I'm not sure I understand the question. There are lots of hyperbolic metrics on your surface, that don't change the group, and lots of presentations, that don't change the metric. What kind of relationship are you looking for?
$endgroup$
– Hempelicious
Jan 10 at 23:09
1
$begingroup$
You are confusing/conflating based homotopy classes (aka elements of $G=pi_1(S,x)$) and free homotopy classes (aka conjugacy classes in $G$). Which do you really mean? If you mean based homotopy classes then the relation of the word length and the hyperbolic length is given by Milnor-Schwarz lemma.
$endgroup$
– Moishe Cohen
Jan 11 at 0:57
$begingroup$
@MoisheCohen isn't it true that free homotopy classes have a one-to-one correspondence with conjugacy classes of $pi_1(S,x)$? And could you expand on the significance of the Milnor-Schwartz lemma in this case?
$endgroup$
– 3891780
Jan 11 at 10:36
1
$begingroup$
@3891780: 1. Yes, this is what I said in my comment. 2. I could, but it is unclear to me what your question really is. Are you talking about based or unbased (free) homotopy classes? Do you know what Milnor-Schwarz lemma is? Do you know about Morse lemma (stability of geodesics)?
$endgroup$
– Moishe Cohen
Jan 11 at 18:06
|
show 4 more comments
1
$begingroup$
In group terms, define in any finitely generated group with given word metric, $c(g)$ as $min_h|hgh^{-1}|$. Then in your setting the length of the geodesic defined by $w$ is comparable to $c(w)$. (All this is unrelated to the choice of relators. Only the choice of generators matters.)
$endgroup$
– YCor
Jan 10 at 19:43
$begingroup$
I'm not sure I understand the question. There are lots of hyperbolic metrics on your surface, that don't change the group, and lots of presentations, that don't change the metric. What kind of relationship are you looking for?
$endgroup$
– Hempelicious
Jan 10 at 23:09
1
$begingroup$
You are confusing/conflating based homotopy classes (aka elements of $G=pi_1(S,x)$) and free homotopy classes (aka conjugacy classes in $G$). Which do you really mean? If you mean based homotopy classes then the relation of the word length and the hyperbolic length is given by Milnor-Schwarz lemma.
$endgroup$
– Moishe Cohen
Jan 11 at 0:57
$begingroup$
@MoisheCohen isn't it true that free homotopy classes have a one-to-one correspondence with conjugacy classes of $pi_1(S,x)$? And could you expand on the significance of the Milnor-Schwartz lemma in this case?
$endgroup$
– 3891780
Jan 11 at 10:36
1
$begingroup$
@3891780: 1. Yes, this is what I said in my comment. 2. I could, but it is unclear to me what your question really is. Are you talking about based or unbased (free) homotopy classes? Do you know what Milnor-Schwarz lemma is? Do you know about Morse lemma (stability of geodesics)?
$endgroup$
– Moishe Cohen
Jan 11 at 18:06
1
1
$begingroup$
In group terms, define in any finitely generated group with given word metric, $c(g)$ as $min_h|hgh^{-1}|$. Then in your setting the length of the geodesic defined by $w$ is comparable to $c(w)$. (All this is unrelated to the choice of relators. Only the choice of generators matters.)
$endgroup$
– YCor
Jan 10 at 19:43
$begingroup$
In group terms, define in any finitely generated group with given word metric, $c(g)$ as $min_h|hgh^{-1}|$. Then in your setting the length of the geodesic defined by $w$ is comparable to $c(w)$. (All this is unrelated to the choice of relators. Only the choice of generators matters.)
$endgroup$
– YCor
Jan 10 at 19:43
$begingroup$
I'm not sure I understand the question. There are lots of hyperbolic metrics on your surface, that don't change the group, and lots of presentations, that don't change the metric. What kind of relationship are you looking for?
$endgroup$
– Hempelicious
Jan 10 at 23:09
$begingroup$
I'm not sure I understand the question. There are lots of hyperbolic metrics on your surface, that don't change the group, and lots of presentations, that don't change the metric. What kind of relationship are you looking for?
$endgroup$
– Hempelicious
Jan 10 at 23:09
1
1
$begingroup$
You are confusing/conflating based homotopy classes (aka elements of $G=pi_1(S,x)$) and free homotopy classes (aka conjugacy classes in $G$). Which do you really mean? If you mean based homotopy classes then the relation of the word length and the hyperbolic length is given by Milnor-Schwarz lemma.
$endgroup$
– Moishe Cohen
Jan 11 at 0:57
$begingroup$
You are confusing/conflating based homotopy classes (aka elements of $G=pi_1(S,x)$) and free homotopy classes (aka conjugacy classes in $G$). Which do you really mean? If you mean based homotopy classes then the relation of the word length and the hyperbolic length is given by Milnor-Schwarz lemma.
$endgroup$
– Moishe Cohen
Jan 11 at 0:57
$begingroup$
@MoisheCohen isn't it true that free homotopy classes have a one-to-one correspondence with conjugacy classes of $pi_1(S,x)$? And could you expand on the significance of the Milnor-Schwartz lemma in this case?
$endgroup$
– 3891780
Jan 11 at 10:36
$begingroup$
@MoisheCohen isn't it true that free homotopy classes have a one-to-one correspondence with conjugacy classes of $pi_1(S,x)$? And could you expand on the significance of the Milnor-Schwartz lemma in this case?
$endgroup$
– 3891780
Jan 11 at 10:36
1
1
$begingroup$
@3891780: 1. Yes, this is what I said in my comment. 2. I could, but it is unclear to me what your question really is. Are you talking about based or unbased (free) homotopy classes? Do you know what Milnor-Schwarz lemma is? Do you know about Morse lemma (stability of geodesics)?
$endgroup$
– Moishe Cohen
Jan 11 at 18:06
$begingroup$
@3891780: 1. Yes, this is what I said in my comment. 2. I could, but it is unclear to me what your question really is. Are you talking about based or unbased (free) homotopy classes? Do you know what Milnor-Schwarz lemma is? Do you know about Morse lemma (stability of geodesics)?
$endgroup$
– Moishe Cohen
Jan 11 at 18:06
|
show 4 more comments
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$begingroup$
In group terms, define in any finitely generated group with given word metric, $c(g)$ as $min_h|hgh^{-1}|$. Then in your setting the length of the geodesic defined by $w$ is comparable to $c(w)$. (All this is unrelated to the choice of relators. Only the choice of generators matters.)
$endgroup$
– YCor
Jan 10 at 19:43
$begingroup$
I'm not sure I understand the question. There are lots of hyperbolic metrics on your surface, that don't change the group, and lots of presentations, that don't change the metric. What kind of relationship are you looking for?
$endgroup$
– Hempelicious
Jan 10 at 23:09
1
$begingroup$
You are confusing/conflating based homotopy classes (aka elements of $G=pi_1(S,x)$) and free homotopy classes (aka conjugacy classes in $G$). Which do you really mean? If you mean based homotopy classes then the relation of the word length and the hyperbolic length is given by Milnor-Schwarz lemma.
$endgroup$
– Moishe Cohen
Jan 11 at 0:57
$begingroup$
@MoisheCohen isn't it true that free homotopy classes have a one-to-one correspondence with conjugacy classes of $pi_1(S,x)$? And could you expand on the significance of the Milnor-Schwartz lemma in this case?
$endgroup$
– 3891780
Jan 11 at 10:36
1
$begingroup$
@3891780: 1. Yes, this is what I said in my comment. 2. I could, but it is unclear to me what your question really is. Are you talking about based or unbased (free) homotopy classes? Do you know what Milnor-Schwarz lemma is? Do you know about Morse lemma (stability of geodesics)?
$endgroup$
– Moishe Cohen
Jan 11 at 18:06