What is a geodesic in Outer space?

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP











up vote
6
down vote

favorite
1












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 $textCV_2$?










share|cite|improve this question



























    up vote
    6
    down vote

    favorite
    1












    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 $textCV_2$?










    share|cite|improve this question

























      up vote
      6
      down vote

      favorite
      1









      up vote
      6
      down vote

      favorite
      1






      1





      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 $textCV_2$?










      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






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Oct 3 at 21:37









      YCor

      25.5k277120




      25.5k277120










      asked Oct 3 at 12:25









      Kim

      31229




      31229




















          4 Answers
          4






          active

          oldest

          votes

















          up vote
          7
          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
            6
            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
              6
              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
                -1
                down vote













                You should consult the oeuvre of Yael Algom-Kfir, in particular "Strongly Contracting Geodesics in Outer Space", whereupon enlightenment will ensue.






                share|cite|improve this answer






















                  Your Answer




                  StackExchange.ifUsing("editor", function ()
                  return StackExchange.using("mathjaxEditing", function ()
                  StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
                  StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
                  );
                  );
                  , "mathjax-editing");

                  StackExchange.ready(function()
                  var channelOptions =
                  tags: "".split(" "),
                  id: "504"
                  ;
                  initTagRenderer("".split(" "), "".split(" "), channelOptions);

                  StackExchange.using("externalEditor", function()
                  // Have to fire editor after snippets, if snippets enabled
                  if (StackExchange.settings.snippets.snippetsEnabled)
                  StackExchange.using("snippets", function()
                  createEditor();
                  );

                  else
                  createEditor();

                  );

                  function createEditor()
                  StackExchange.prepareEditor(
                  heartbeatType: 'answer',
                  convertImagesToLinks: true,
                  noModals: false,
                  showLowRepImageUploadWarning: true,
                  reputationToPostImages: 10,
                  bindNavPrevention: true,
                  postfix: "",
                  noCode: true, onDemand: true,
                  discardSelector: ".discard-answer"
                  ,immediatelyShowMarkdownHelp:true
                  );



                  );













                   

                  draft saved


                  draft discarded


















                  StackExchange.ready(
                  function ()
                  StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f311948%2fwhat-is-a-geodesic-in-outer-space%23new-answer', 'question_page');

                  );

                  Post as a guest






























                  4 Answers
                  4






                  active

                  oldest

                  votes








                  4 Answers
                  4






                  active

                  oldest

                  votes









                  active

                  oldest

                  votes






                  active

                  oldest

                  votes








                  up vote
                  7
                  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
                    7
                    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
                      7
                      down vote










                      up vote
                      7
                      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 Oct 3 at 23:47









                      Lee Mosher

                      12.8k22663




                      12.8k22663




















                          up vote
                          6
                          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
                            6
                            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
                              6
                              down vote










                              up vote
                              6
                              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 Oct 3 at 19:06









                              Andy Putman

                              30.3k5130208




                              30.3k5130208




















                                  up vote
                                  6
                                  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
                                    6
                                    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
                                      6
                                      down vote










                                      up vote
                                      6
                                      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 Oct 3 at 20:53









                                      Harry Richman

                                      926517




                                      926517




















                                          up vote
                                          -1
                                          down vote













                                          You should consult the oeuvre of Yael Algom-Kfir, in particular "Strongly Contracting Geodesics in Outer Space", whereupon enlightenment will ensue.






                                          share|cite|improve this answer


























                                            up vote
                                            -1
                                            down vote













                                            You should consult the oeuvre of Yael Algom-Kfir, in particular "Strongly Contracting Geodesics in Outer Space", whereupon enlightenment will ensue.






                                            share|cite|improve this answer
























                                              up vote
                                              -1
                                              down vote










                                              up vote
                                              -1
                                              down vote









                                              You should consult the oeuvre of Yael Algom-Kfir, in particular "Strongly Contracting Geodesics in Outer Space", whereupon enlightenment will ensue.






                                              share|cite|improve this answer














                                              You should consult the oeuvre of Yael Algom-Kfir, in particular "Strongly Contracting Geodesics in Outer Space", whereupon enlightenment will ensue.







                                              share|cite|improve this answer














                                              share|cite|improve this answer



                                              share|cite|improve this answer








                                              edited Oct 4 at 4:06









                                              Alex M.

                                              2,36831531




                                              2,36831531










                                              answered Oct 3 at 18:39









                                              Igor Rivin

                                              78.1k8111304




                                              78.1k8111304



























                                                   

                                                  draft saved


                                                  draft discarded















































                                                   


                                                  draft saved


                                                  draft discarded














                                                  StackExchange.ready(
                                                  function ()
                                                  StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f311948%2fwhat-is-a-geodesic-in-outer-space%23new-answer', 'question_page');

                                                  );

                                                  Post as a guest













































































                                                  Popular posts from this blog

                                                  How to check contact read email or not when send email to Individual?

                                                  Displaying single band from multi-band raster using QGIS

                                                  How many registers does an x86_64 CPU actually have?