Are free groups and free actions related?












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Is there a connection between free groups and free actions, or is it that their names just happen to be the same? I'm studying groups theory at the moment, and haven't found any relation between the two, and I find it a bit odd that the names are so similar.










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  • 7




    $begingroup$
    Free actions are free objects in the category of $G$-sets. This is probably not where the name comes from, though. They both mean free as in unconstrained.
    $endgroup$
    – Qiaochu Yuan
    Dec 25 '18 at 20:41










  • $begingroup$
    "Free" is a common word, like "torsion-free" etc. They are not necessarily all related.
    $endgroup$
    – Dietrich Burde
    Dec 25 '18 at 21:08












  • $begingroup$
    @QiaochuYuan that should be an answer.
    $endgroup$
    – freakish
    Dec 25 '18 at 21:38










  • $begingroup$
    @QiaochuYuan I'll take that as an answer, Thank you!
    $endgroup$
    – Rei Henigman
    Dec 26 '18 at 19:12






  • 1




    $begingroup$
    Free groups are exactly the groups which act freely on trees(of course this is a very special case of a free action).
    $endgroup$
    – Paul Plummer
    Dec 27 '18 at 17:29
















0












$begingroup$


Is there a connection between free groups and free actions, or is it that their names just happen to be the same? I'm studying groups theory at the moment, and haven't found any relation between the two, and I find it a bit odd that the names are so similar.










share|cite|improve this question









$endgroup$








  • 7




    $begingroup$
    Free actions are free objects in the category of $G$-sets. This is probably not where the name comes from, though. They both mean free as in unconstrained.
    $endgroup$
    – Qiaochu Yuan
    Dec 25 '18 at 20:41










  • $begingroup$
    "Free" is a common word, like "torsion-free" etc. They are not necessarily all related.
    $endgroup$
    – Dietrich Burde
    Dec 25 '18 at 21:08












  • $begingroup$
    @QiaochuYuan that should be an answer.
    $endgroup$
    – freakish
    Dec 25 '18 at 21:38










  • $begingroup$
    @QiaochuYuan I'll take that as an answer, Thank you!
    $endgroup$
    – Rei Henigman
    Dec 26 '18 at 19:12






  • 1




    $begingroup$
    Free groups are exactly the groups which act freely on trees(of course this is a very special case of a free action).
    $endgroup$
    – Paul Plummer
    Dec 27 '18 at 17:29














0












0








0





$begingroup$


Is there a connection between free groups and free actions, or is it that their names just happen to be the same? I'm studying groups theory at the moment, and haven't found any relation between the two, and I find it a bit odd that the names are so similar.










share|cite|improve this question









$endgroup$




Is there a connection between free groups and free actions, or is it that their names just happen to be the same? I'm studying groups theory at the moment, and haven't found any relation between the two, and I find it a bit odd that the names are so similar.







group-theory free-groups






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share|cite|improve this question










asked Dec 25 '18 at 20:38









Rei HenigmanRei Henigman

717




717








  • 7




    $begingroup$
    Free actions are free objects in the category of $G$-sets. This is probably not where the name comes from, though. They both mean free as in unconstrained.
    $endgroup$
    – Qiaochu Yuan
    Dec 25 '18 at 20:41










  • $begingroup$
    "Free" is a common word, like "torsion-free" etc. They are not necessarily all related.
    $endgroup$
    – Dietrich Burde
    Dec 25 '18 at 21:08












  • $begingroup$
    @QiaochuYuan that should be an answer.
    $endgroup$
    – freakish
    Dec 25 '18 at 21:38










  • $begingroup$
    @QiaochuYuan I'll take that as an answer, Thank you!
    $endgroup$
    – Rei Henigman
    Dec 26 '18 at 19:12






  • 1




    $begingroup$
    Free groups are exactly the groups which act freely on trees(of course this is a very special case of a free action).
    $endgroup$
    – Paul Plummer
    Dec 27 '18 at 17:29














  • 7




    $begingroup$
    Free actions are free objects in the category of $G$-sets. This is probably not where the name comes from, though. They both mean free as in unconstrained.
    $endgroup$
    – Qiaochu Yuan
    Dec 25 '18 at 20:41










  • $begingroup$
    "Free" is a common word, like "torsion-free" etc. They are not necessarily all related.
    $endgroup$
    – Dietrich Burde
    Dec 25 '18 at 21:08












  • $begingroup$
    @QiaochuYuan that should be an answer.
    $endgroup$
    – freakish
    Dec 25 '18 at 21:38










  • $begingroup$
    @QiaochuYuan I'll take that as an answer, Thank you!
    $endgroup$
    – Rei Henigman
    Dec 26 '18 at 19:12






  • 1




    $begingroup$
    Free groups are exactly the groups which act freely on trees(of course this is a very special case of a free action).
    $endgroup$
    – Paul Plummer
    Dec 27 '18 at 17:29








7




7




$begingroup$
Free actions are free objects in the category of $G$-sets. This is probably not where the name comes from, though. They both mean free as in unconstrained.
$endgroup$
– Qiaochu Yuan
Dec 25 '18 at 20:41




$begingroup$
Free actions are free objects in the category of $G$-sets. This is probably not where the name comes from, though. They both mean free as in unconstrained.
$endgroup$
– Qiaochu Yuan
Dec 25 '18 at 20:41












$begingroup$
"Free" is a common word, like "torsion-free" etc. They are not necessarily all related.
$endgroup$
– Dietrich Burde
Dec 25 '18 at 21:08






$begingroup$
"Free" is a common word, like "torsion-free" etc. They are not necessarily all related.
$endgroup$
– Dietrich Burde
Dec 25 '18 at 21:08














$begingroup$
@QiaochuYuan that should be an answer.
$endgroup$
– freakish
Dec 25 '18 at 21:38




$begingroup$
@QiaochuYuan that should be an answer.
$endgroup$
– freakish
Dec 25 '18 at 21:38












$begingroup$
@QiaochuYuan I'll take that as an answer, Thank you!
$endgroup$
– Rei Henigman
Dec 26 '18 at 19:12




$begingroup$
@QiaochuYuan I'll take that as an answer, Thank you!
$endgroup$
– Rei Henigman
Dec 26 '18 at 19:12




1




1




$begingroup$
Free groups are exactly the groups which act freely on trees(of course this is a very special case of a free action).
$endgroup$
– Paul Plummer
Dec 27 '18 at 17:29




$begingroup$
Free groups are exactly the groups which act freely on trees(of course this is a very special case of a free action).
$endgroup$
– Paul Plummer
Dec 27 '18 at 17:29










1 Answer
1






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$begingroup$

Free $G$-actions are free objects in the category of G-sets.



More precisely, a free object on a set $I$ in this category is the same as a free $G$-set $X$ endowed with a map $Ito G$ meeting once each orbit; the obvious way to produce it is just considering $X=Gtimes I$ with action $gcdot (h,i)=(gh,i)$ and the map $Ito X$, $imapsto (1,i)$.



For a free action on a set $I$, the maps from various sets to $X$ satisfying the given property are the free generating families of $X$ as $G$-set.






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    $begingroup$

    Free $G$-actions are free objects in the category of G-sets.



    More precisely, a free object on a set $I$ in this category is the same as a free $G$-set $X$ endowed with a map $Ito G$ meeting once each orbit; the obvious way to produce it is just considering $X=Gtimes I$ with action $gcdot (h,i)=(gh,i)$ and the map $Ito X$, $imapsto (1,i)$.



    For a free action on a set $I$, the maps from various sets to $X$ satisfying the given property are the free generating families of $X$ as $G$-set.






    share|cite|improve this answer











    $endgroup$


















      2












      $begingroup$

      Free $G$-actions are free objects in the category of G-sets.



      More precisely, a free object on a set $I$ in this category is the same as a free $G$-set $X$ endowed with a map $Ito G$ meeting once each orbit; the obvious way to produce it is just considering $X=Gtimes I$ with action $gcdot (h,i)=(gh,i)$ and the map $Ito X$, $imapsto (1,i)$.



      For a free action on a set $I$, the maps from various sets to $X$ satisfying the given property are the free generating families of $X$ as $G$-set.






      share|cite|improve this answer











      $endgroup$
















        2












        2








        2





        $begingroup$

        Free $G$-actions are free objects in the category of G-sets.



        More precisely, a free object on a set $I$ in this category is the same as a free $G$-set $X$ endowed with a map $Ito G$ meeting once each orbit; the obvious way to produce it is just considering $X=Gtimes I$ with action $gcdot (h,i)=(gh,i)$ and the map $Ito X$, $imapsto (1,i)$.



        For a free action on a set $I$, the maps from various sets to $X$ satisfying the given property are the free generating families of $X$ as $G$-set.






        share|cite|improve this answer











        $endgroup$



        Free $G$-actions are free objects in the category of G-sets.



        More precisely, a free object on a set $I$ in this category is the same as a free $G$-set $X$ endowed with a map $Ito G$ meeting once each orbit; the obvious way to produce it is just considering $X=Gtimes I$ with action $gcdot (h,i)=(gh,i)$ and the map $Ito X$, $imapsto (1,i)$.



        For a free action on a set $I$, the maps from various sets to $X$ satisfying the given property are the free generating families of $X$ as $G$-set.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        answered Jan 11 at 3:09


























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