What is a geodesic in Outer space?

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The Culler-Vogtmann Outer space $textCV_n$ is an analogue of Teichmuller space for the group $textOut(F_n)$.



Is there any notion of a geodesic path in $textCV_n$? Are there different competing definitions of geodesic?



If so, what would be a simple example of a geodesic path vs. a non-geodesic one, say on $textCV_2$?






gr.group-theory gt.geometric-topology geometric-group-theory teichmuller-theory free-groups







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    up vote
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    The Culler-Vogtmann Outer space $textCV_n$ is an analogue of Teichmuller space for the group $textOut(F_n)$.



    Is there any notion of a geodesic path in $textCV_n$? Are there different competing definitions of geodesic?



    If so, what would be a simple example of a geodesic path vs. a non-geodesic one, say on $textCV_2$?






    gr.group-theory gt.geometric-topology geometric-group-theory teichmuller-theory free-groups







    share|cite|improve this question

























      up vote
      6
      down vote

      favorite









      up vote
      6
      down vote

      favorite











      The Culler-Vogtmann Outer space $textCV_n$ is an analogue of Teichmuller space for the group $textOut(F_n)$.



      Is there any notion of a geodesic path in $textCV_n$? Are there different competing definitions of geodesic?



      If so, what would be a simple example of a geodesic path vs. a non-geodesic one, say on $textCV_2$?






      gr.group-theory gt.geometric-topology geometric-group-theory teichmuller-theory free-groups







      share|cite|improve this question















      The Culler-Vogtmann Outer space $textCV_n$ is an analogue of Teichmuller space for the group $textOut(F_n)$.



      Is there any notion of a geodesic path in $textCV_n$? Are there different competing definitions of geodesic?



      If so, what would be a simple example of a geodesic path vs. a non-geodesic one, say on $textCV_2$?






      gr.group-theory gt.geometric-topology geometric-group-theory teichmuller-theory free-groups




      gr.group-theory gt.geometric-topology geometric-group-theory teichmuller-theory free-groups






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      edited 4 hours ago









      YCor

      25.4k277120




      25.4k277120










      asked 13 hours ago









      Kim

      28728




      28728




















          4 Answers
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          To talk about geodesics, you need a notion of distance. For outer space, something strange happens: there is a natural notion of distance (called the "Lipshitz metric"), but it is not symmetric. In other words, there exist points $x$ and $y$ in Outer space such that $d(x,y)$ and $d(y,x)$ are different! Nonetheless, one can still talk about geodesics.



          For an introduction to this circle of ideas, I recommend Bestvina's Park City notes:



          Bestvina, Mladen,
          Geometry of outer space. Geometric group theory, 173–206,
          IAS/Park City Math. Ser., 21, Amer. Math. Soc., Providence, RI, 2014.



          The whole set of notes is useful, but Lecture 3 is where the distance function is defined.






          share|cite|improve this answer



























            up vote
            3
            down vote













            I would recommend looking at Karen Vogtmann's survey article On the geometry of Outer space, in the Bulletin of the AMS (and available online).
            The Lipschitz metric is defined and discussed in Section 3, and
            Section 5 discusses geodesics in this metric.



            In particular, the following excerpt taken from pages 37-38 contains an example of two directed geodesics in $textCV_2$ between the same two endpoints in adjacent cells.
            (As mentioned in @andyputman's answer, the Lipschitz metric is not symmetric so geodesics generally depend on the direction of travel.)
            In this example, any path between the two endpoints which crosses
            the boundary line between the cells
            at a different point would not be geodesic.








            share|cite|improve this answer



























              up vote
              3
              down vote













              Besides the geodesic paths of the asymmetric metric $d(cdot,cdot)$ that are mentioned in other answers (namely paths such that $d(gamma(s),gamma(t)) = t-s$ if $s le t$), there is another class of paths with many uses known as Stallings fold paths. You can see some discusions of them in the outer space context, with applications, in these lecture notes of Bestvina, these notes of Kapovich and Myasnikov, and this issue of the AMS Memoirs by Handel and myself.






              share|cite|improve this answer



























                up vote
                -3
                down vote













                You should go to http://arxiv.org, and consult the oeuvre of Yael Algom-Kfir, whereupon enlightenment will ensue.






                share|cite|improve this answer
















                • 6




                  There are 11 papers by that author on arXiv, so that's some hundreds of pages of work to look through. Could you be any more specific?
                  – Nate Eldredge
                  7 hours ago










                • The very first paper in the chronological sequence defines them (strongly contracting, etc).
                  – Igor Rivin
                  7 hours ago










                • Wow, two downvotes?
                  – Igor Rivin
                  7 hours ago










                • @NateEldredge I thought it was obvious that any paper with "geodesics in outer space" in the title would be the one you would look at first.
                  – Igor Rivin
                  7 hours ago










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                4 Answers
                4






                active

                oldest

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                4 Answers
                4






                active

                oldest

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                active

                oldest

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                active

                oldest

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                up vote
                5
                down vote













                To talk about geodesics, you need a notion of distance. For outer space, something strange happens: there is a natural notion of distance (called the "Lipshitz metric"), but it is not symmetric. In other words, there exist points $x$ and $y$ in Outer space such that $d(x,y)$ and $d(y,x)$ are different! Nonetheless, one can still talk about geodesics.



                For an introduction to this circle of ideas, I recommend Bestvina's Park City notes:



                Bestvina, Mladen,
                Geometry of outer space. Geometric group theory, 173–206,
                IAS/Park City Math. Ser., 21, Amer. Math. Soc., Providence, RI, 2014.



                The whole set of notes is useful, but Lecture 3 is where the distance function is defined.






                share|cite|improve this answer
























                  up vote
                  5
                  down vote













                  To talk about geodesics, you need a notion of distance. For outer space, something strange happens: there is a natural notion of distance (called the "Lipshitz metric"), but it is not symmetric. In other words, there exist points $x$ and $y$ in Outer space such that $d(x,y)$ and $d(y,x)$ are different! Nonetheless, one can still talk about geodesics.



                  For an introduction to this circle of ideas, I recommend Bestvina's Park City notes:



                  Bestvina, Mladen,
                  Geometry of outer space. Geometric group theory, 173–206,
                  IAS/Park City Math. Ser., 21, Amer. Math. Soc., Providence, RI, 2014.



                  The whole set of notes is useful, but Lecture 3 is where the distance function is defined.






                  share|cite|improve this answer






















                    up vote
                    5
                    down vote










                    up vote
                    5
                    down vote









                    To talk about geodesics, you need a notion of distance. For outer space, something strange happens: there is a natural notion of distance (called the "Lipshitz metric"), but it is not symmetric. In other words, there exist points $x$ and $y$ in Outer space such that $d(x,y)$ and $d(y,x)$ are different! Nonetheless, one can still talk about geodesics.



                    For an introduction to this circle of ideas, I recommend Bestvina's Park City notes:



                    Bestvina, Mladen,
                    Geometry of outer space. Geometric group theory, 173–206,
                    IAS/Park City Math. Ser., 21, Amer. Math. Soc., Providence, RI, 2014.



                    The whole set of notes is useful, but Lecture 3 is where the distance function is defined.






                    share|cite|improve this answer












                    To talk about geodesics, you need a notion of distance. For outer space, something strange happens: there is a natural notion of distance (called the "Lipshitz metric"), but it is not symmetric. In other words, there exist points $x$ and $y$ in Outer space such that $d(x,y)$ and $d(y,x)$ are different! Nonetheless, one can still talk about geodesics.



                    For an introduction to this circle of ideas, I recommend Bestvina's Park City notes:



                    Bestvina, Mladen,
                    Geometry of outer space. Geometric group theory, 173–206,
                    IAS/Park City Math. Ser., 21, Amer. Math. Soc., Providence, RI, 2014.



                    The whole set of notes is useful, but Lecture 3 is where the distance function is defined.







                    share|cite|improve this answer












                    share|cite|improve this answer



                    share|cite|improve this answer










                    answered 7 hours ago









                    Andy Putman

                    30.3k5130208




                    30.3k5130208




















                        up vote
                        3
                        down vote













                        I would recommend looking at Karen Vogtmann's survey article On the geometry of Outer space, in the Bulletin of the AMS (and available online).
                        The Lipschitz metric is defined and discussed in Section 3, and
                        Section 5 discusses geodesics in this metric.



                        In particular, the following excerpt taken from pages 37-38 contains an example of two directed geodesics in $textCV_2$ between the same two endpoints in adjacent cells.
                        (As mentioned in @andyputman's answer, the Lipschitz metric is not symmetric so geodesics generally depend on the direction of travel.)
                        In this example, any path between the two endpoints which crosses
                        the boundary line between the cells
                        at a different point would not be geodesic.








                        share|cite|improve this answer
























                          up vote
                          3
                          down vote













                          I would recommend looking at Karen Vogtmann's survey article On the geometry of Outer space, in the Bulletin of the AMS (and available online).
                          The Lipschitz metric is defined and discussed in Section 3, and
                          Section 5 discusses geodesics in this metric.



                          In particular, the following excerpt taken from pages 37-38 contains an example of two directed geodesics in $textCV_2$ between the same two endpoints in adjacent cells.
                          (As mentioned in @andyputman's answer, the Lipschitz metric is not symmetric so geodesics generally depend on the direction of travel.)
                          In this example, any path between the two endpoints which crosses
                          the boundary line between the cells
                          at a different point would not be geodesic.








                          share|cite|improve this answer






















                            up vote
                            3
                            down vote










                            up vote
                            3
                            down vote









                            I would recommend looking at Karen Vogtmann's survey article On the geometry of Outer space, in the Bulletin of the AMS (and available online).
                            The Lipschitz metric is defined and discussed in Section 3, and
                            Section 5 discusses geodesics in this metric.



                            In particular, the following excerpt taken from pages 37-38 contains an example of two directed geodesics in $textCV_2$ between the same two endpoints in adjacent cells.
                            (As mentioned in @andyputman's answer, the Lipschitz metric is not symmetric so geodesics generally depend on the direction of travel.)
                            In this example, any path between the two endpoints which crosses
                            the boundary line between the cells
                            at a different point would not be geodesic.








                            share|cite|improve this answer












                            I would recommend looking at Karen Vogtmann's survey article On the geometry of Outer space, in the Bulletin of the AMS (and available online).
                            The Lipschitz metric is defined and discussed in Section 3, and
                            Section 5 discusses geodesics in this metric.



                            In particular, the following excerpt taken from pages 37-38 contains an example of two directed geodesics in $textCV_2$ between the same two endpoints in adjacent cells.
                            (As mentioned in @andyputman's answer, the Lipschitz metric is not symmetric so geodesics generally depend on the direction of travel.)
                            In this example, any path between the two endpoints which crosses
                            the boundary line between the cells
                            at a different point would not be geodesic.









                            share|cite|improve this answer












                            share|cite|improve this answer



                            share|cite|improve this answer










                            answered 5 hours ago









                            Harry Richman

                            896517




                            896517




















                                up vote
                                3
                                down vote













                                Besides the geodesic paths of the asymmetric metric $d(cdot,cdot)$ that are mentioned in other answers (namely paths such that $d(gamma(s),gamma(t)) = t-s$ if $s le t$), there is another class of paths with many uses known as Stallings fold paths. You can see some discusions of them in the outer space context, with applications, in these lecture notes of Bestvina, these notes of Kapovich and Myasnikov, and this issue of the AMS Memoirs by Handel and myself.






                                share|cite|improve this answer
























                                  up vote
                                  3
                                  down vote













                                  Besides the geodesic paths of the asymmetric metric $d(cdot,cdot)$ that are mentioned in other answers (namely paths such that $d(gamma(s),gamma(t)) = t-s$ if $s le t$), there is another class of paths with many uses known as Stallings fold paths. You can see some discusions of them in the outer space context, with applications, in these lecture notes of Bestvina, these notes of Kapovich and Myasnikov, and this issue of the AMS Memoirs by Handel and myself.






                                  share|cite|improve this answer






















                                    up vote
                                    3
                                    down vote










                                    up vote
                                    3
                                    down vote









                                    Besides the geodesic paths of the asymmetric metric $d(cdot,cdot)$ that are mentioned in other answers (namely paths such that $d(gamma(s),gamma(t)) = t-s$ if $s le t$), there is another class of paths with many uses known as Stallings fold paths. You can see some discusions of them in the outer space context, with applications, in these lecture notes of Bestvina, these notes of Kapovich and Myasnikov, and this issue of the AMS Memoirs by Handel and myself.






                                    share|cite|improve this answer












                                    Besides the geodesic paths of the asymmetric metric $d(cdot,cdot)$ that are mentioned in other answers (namely paths such that $d(gamma(s),gamma(t)) = t-s$ if $s le t$), there is another class of paths with many uses known as Stallings fold paths. You can see some discusions of them in the outer space context, with applications, in these lecture notes of Bestvina, these notes of Kapovich and Myasnikov, and this issue of the AMS Memoirs by Handel and myself.







                                    share|cite|improve this answer












                                    share|cite|improve this answer



                                    share|cite|improve this answer










                                    answered 2 hours ago









                                    Lee Mosher

                                    12.7k22663




                                    12.7k22663




















                                        up vote
                                        -3
                                        down vote













                                        You should go to http://arxiv.org, and consult the oeuvre of Yael Algom-Kfir, whereupon enlightenment will ensue.






                                        share|cite|improve this answer
















                                        • 6




                                          There are 11 papers by that author on arXiv, so that's some hundreds of pages of work to look through. Could you be any more specific?
                                          – Nate Eldredge
                                          7 hours ago










                                        • The very first paper in the chronological sequence defines them (strongly contracting, etc).
                                          – Igor Rivin
                                          7 hours ago










                                        • Wow, two downvotes?
                                          – Igor Rivin
                                          7 hours ago










                                        • @NateEldredge I thought it was obvious that any paper with "geodesics in outer space" in the title would be the one you would look at first.
                                          – Igor Rivin
                                          7 hours ago














                                        up vote
                                        -3
                                        down vote













                                        You should go to http://arxiv.org, and consult the oeuvre of Yael Algom-Kfir, whereupon enlightenment will ensue.






                                        share|cite|improve this answer
















                                        • 6




                                          There are 11 papers by that author on arXiv, so that's some hundreds of pages of work to look through. Could you be any more specific?
                                          – Nate Eldredge
                                          7 hours ago










                                        • The very first paper in the chronological sequence defines them (strongly contracting, etc).
                                          – Igor Rivin
                                          7 hours ago










                                        • Wow, two downvotes?
                                          – Igor Rivin
                                          7 hours ago










                                        • @NateEldredge I thought it was obvious that any paper with "geodesics in outer space" in the title would be the one you would look at first.
                                          – Igor Rivin
                                          7 hours ago












                                        up vote
                                        -3
                                        down vote










                                        up vote
                                        -3
                                        down vote









                                        You should go to http://arxiv.org, and consult the oeuvre of Yael Algom-Kfir, whereupon enlightenment will ensue.






                                        share|cite|improve this answer












                                        You should go to http://arxiv.org, and consult the oeuvre of Yael Algom-Kfir, whereupon enlightenment will ensue.







                                        share|cite|improve this answer












                                        share|cite|improve this answer



                                        share|cite|improve this answer










                                        answered 7 hours ago









                                        Igor Rivin

                                        77.9k8111304




                                        77.9k8111304







                                        • 6




                                          There are 11 papers by that author on arXiv, so that's some hundreds of pages of work to look through. Could you be any more specific?
                                          – Nate Eldredge
                                          7 hours ago










                                        • The very first paper in the chronological sequence defines them (strongly contracting, etc).
                                          – Igor Rivin
                                          7 hours ago










                                        • Wow, two downvotes?
                                          – Igor Rivin
                                          7 hours ago










                                        • @NateEldredge I thought it was obvious that any paper with "geodesics in outer space" in the title would be the one you would look at first.
                                          – Igor Rivin
                                          7 hours ago












                                        • 6




                                          There are 11 papers by that author on arXiv, so that's some hundreds of pages of work to look through. Could you be any more specific?
                                          – Nate Eldredge
                                          7 hours ago










                                        • The very first paper in the chronological sequence defines them (strongly contracting, etc).
                                          – Igor Rivin
                                          7 hours ago










                                        • Wow, two downvotes?
                                          – Igor Rivin
                                          7 hours ago










                                        • @NateEldredge I thought it was obvious that any paper with "geodesics in outer space" in the title would be the one you would look at first.
                                          – Igor Rivin
                                          7 hours ago







                                        6




                                        6




                                        There are 11 papers by that author on arXiv, so that's some hundreds of pages of work to look through. Could you be any more specific?
                                        – Nate Eldredge
                                        7 hours ago




                                        There are 11 papers by that author on arXiv, so that's some hundreds of pages of work to look through. Could you be any more specific?
                                        – Nate Eldredge
                                        7 hours ago












                                        The very first paper in the chronological sequence defines them (strongly contracting, etc).
                                        – Igor Rivin
                                        7 hours ago




                                        The very first paper in the chronological sequence defines them (strongly contracting, etc).
                                        – Igor Rivin
                                        7 hours ago












                                        Wow, two downvotes?
                                        – Igor Rivin
                                        7 hours ago




                                        Wow, two downvotes?
                                        – Igor Rivin
                                        7 hours ago












                                        @NateEldredge I thought it was obvious that any paper with "geodesics in outer space" in the title would be the one you would look at first.
                                        – Igor Rivin
                                        7 hours ago




                                        @NateEldredge I thought it was obvious that any paper with "geodesics in outer space" in the title would be the one you would look at first.
                                        – Igor Rivin
                                        7 hours ago

















                                         

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