Groups acting on trees












1












$begingroup$


Assume that X is a tree such that every vertex has infinite degree. And a discrete group G acts on this tree properly (with finite stabilizers) and transitively. Is it true that G contains a non- abelian free subgroup?






gr.group-theory







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$endgroup$












  • $begingroup$
    Properly implies finite stabilizers
    $endgroup$
    – YCor
    1 hour ago
















1












$begingroup$


Assume that X is a tree such that every vertex has infinite degree. And a discrete group G acts on this tree properly (with finite stabilizers) and transitively. Is it true that G contains a non- abelian free subgroup?






gr.group-theory







share|cite|improve this question









$endgroup$












  • $begingroup$
    Properly implies finite stabilizers
    $endgroup$
    – YCor
    1 hour ago














1












1








1





$begingroup$


Assume that X is a tree such that every vertex has infinite degree. And a discrete group G acts on this tree properly (with finite stabilizers) and transitively. Is it true that G contains a non- abelian free subgroup?






gr.group-theory







share|cite|improve this question









$endgroup$




Assume that X is a tree such that every vertex has infinite degree. And a discrete group G acts on this tree properly (with finite stabilizers) and transitively. Is it true that G contains a non- abelian free subgroup?






gr.group-theory




gr.group-theory






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share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked 1 hour ago









Maria GerasimovaMaria Gerasimova

1967




1967












  • $begingroup$
    Properly implies finite stabilizers
    $endgroup$
    – YCor
    1 hour ago


















  • $begingroup$
    Properly implies finite stabilizers
    $endgroup$
    – YCor
    1 hour ago
















$begingroup$
Properly implies finite stabilizers
$endgroup$
– YCor
1 hour ago




$begingroup$
Properly implies finite stabilizers
$endgroup$
– YCor
1 hour ago










1 Answer
1






active

oldest

votes


















3












$begingroup$

Yes.



You're assuming more than what's necessary.



For an isometric group action on a Gromov-hyperbolic space, you have 5 possibilities (Gromov's classification):




  • (a) bounded orbits

  • (b) horocyclic (fixes a unique point at infinity, no loxodromic element; preserves "horospheres")

  • (c) axial (preserves an axis, on which some element acts loxodromically)

  • (d) focal (fixes a unique point at infinity, existence of a loxodromic element)

  • (e) general type (= other): implies the existence of a non-abelian free subgroup acting metrically properly.


(a), (b), (c) are clearly ruled out in your setting (transitivity, and valency; valency $ge 3$ would be enough). (d) is ruled out because discrete groups have no metrically proper focal action at all. So (e) holds.



The case of actions on trees is an useful motivating baby case in the above "classification"; all cases can already occur.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Thank you! And if I assume that stabilizers are amenable, will it be the same?
    $endgroup$
    – Maria Gerasimova
    33 mins ago










  • $begingroup$
    @MariaGerasimova you have to discard the focal case. Writing, for instance, the solvable group $mathbf{Z}wrmathbf{Z}$ as an ascending HNN extension (w.r.t. an endomorphism with image of finite index), you make it act vertex-transitively on a tree with infinite valency (with abelian stabilizers). So you need to explicitly exclude the case of a fixed point at infinity.
    $endgroup$
    – YCor
    17 mins ago













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1 Answer
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1 Answer
1






active

oldest

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active

oldest

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active

oldest

votes









3












$begingroup$

Yes.



You're assuming more than what's necessary.



For an isometric group action on a Gromov-hyperbolic space, you have 5 possibilities (Gromov's classification):




  • (a) bounded orbits

  • (b) horocyclic (fixes a unique point at infinity, no loxodromic element; preserves "horospheres")

  • (c) axial (preserves an axis, on which some element acts loxodromically)

  • (d) focal (fixes a unique point at infinity, existence of a loxodromic element)

  • (e) general type (= other): implies the existence of a non-abelian free subgroup acting metrically properly.


(a), (b), (c) are clearly ruled out in your setting (transitivity, and valency; valency $ge 3$ would be enough). (d) is ruled out because discrete groups have no metrically proper focal action at all. So (e) holds.



The case of actions on trees is an useful motivating baby case in the above "classification"; all cases can already occur.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Thank you! And if I assume that stabilizers are amenable, will it be the same?
    $endgroup$
    – Maria Gerasimova
    33 mins ago










  • $begingroup$
    @MariaGerasimova you have to discard the focal case. Writing, for instance, the solvable group $mathbf{Z}wrmathbf{Z}$ as an ascending HNN extension (w.r.t. an endomorphism with image of finite index), you make it act vertex-transitively on a tree with infinite valency (with abelian stabilizers). So you need to explicitly exclude the case of a fixed point at infinity.
    $endgroup$
    – YCor
    17 mins ago


















3












$begingroup$

Yes.



You're assuming more than what's necessary.



For an isometric group action on a Gromov-hyperbolic space, you have 5 possibilities (Gromov's classification):




  • (a) bounded orbits

  • (b) horocyclic (fixes a unique point at infinity, no loxodromic element; preserves "horospheres")

  • (c) axial (preserves an axis, on which some element acts loxodromically)

  • (d) focal (fixes a unique point at infinity, existence of a loxodromic element)

  • (e) general type (= other): implies the existence of a non-abelian free subgroup acting metrically properly.


(a), (b), (c) are clearly ruled out in your setting (transitivity, and valency; valency $ge 3$ would be enough). (d) is ruled out because discrete groups have no metrically proper focal action at all. So (e) holds.



The case of actions on trees is an useful motivating baby case in the above "classification"; all cases can already occur.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Thank you! And if I assume that stabilizers are amenable, will it be the same?
    $endgroup$
    – Maria Gerasimova
    33 mins ago










  • $begingroup$
    @MariaGerasimova you have to discard the focal case. Writing, for instance, the solvable group $mathbf{Z}wrmathbf{Z}$ as an ascending HNN extension (w.r.t. an endomorphism with image of finite index), you make it act vertex-transitively on a tree with infinite valency (with abelian stabilizers). So you need to explicitly exclude the case of a fixed point at infinity.
    $endgroup$
    – YCor
    17 mins ago
















3












3








3





$begingroup$

Yes.



You're assuming more than what's necessary.



For an isometric group action on a Gromov-hyperbolic space, you have 5 possibilities (Gromov's classification):




  • (a) bounded orbits

  • (b) horocyclic (fixes a unique point at infinity, no loxodromic element; preserves "horospheres")

  • (c) axial (preserves an axis, on which some element acts loxodromically)

  • (d) focal (fixes a unique point at infinity, existence of a loxodromic element)

  • (e) general type (= other): implies the existence of a non-abelian free subgroup acting metrically properly.


(a), (b), (c) are clearly ruled out in your setting (transitivity, and valency; valency $ge 3$ would be enough). (d) is ruled out because discrete groups have no metrically proper focal action at all. So (e) holds.



The case of actions on trees is an useful motivating baby case in the above "classification"; all cases can already occur.






share|cite|improve this answer









$endgroup$



Yes.



You're assuming more than what's necessary.



For an isometric group action on a Gromov-hyperbolic space, you have 5 possibilities (Gromov's classification):




  • (a) bounded orbits

  • (b) horocyclic (fixes a unique point at infinity, no loxodromic element; preserves "horospheres")

  • (c) axial (preserves an axis, on which some element acts loxodromically)

  • (d) focal (fixes a unique point at infinity, existence of a loxodromic element)

  • (e) general type (= other): implies the existence of a non-abelian free subgroup acting metrically properly.


(a), (b), (c) are clearly ruled out in your setting (transitivity, and valency; valency $ge 3$ would be enough). (d) is ruled out because discrete groups have no metrically proper focal action at all. So (e) holds.



The case of actions on trees is an useful motivating baby case in the above "classification"; all cases can already occur.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered 58 mins ago









YCorYCor

27.8k482134




27.8k482134












  • $begingroup$
    Thank you! And if I assume that stabilizers are amenable, will it be the same?
    $endgroup$
    – Maria Gerasimova
    33 mins ago










  • $begingroup$
    @MariaGerasimova you have to discard the focal case. Writing, for instance, the solvable group $mathbf{Z}wrmathbf{Z}$ as an ascending HNN extension (w.r.t. an endomorphism with image of finite index), you make it act vertex-transitively on a tree with infinite valency (with abelian stabilizers). So you need to explicitly exclude the case of a fixed point at infinity.
    $endgroup$
    – YCor
    17 mins ago




















  • $begingroup$
    Thank you! And if I assume that stabilizers are amenable, will it be the same?
    $endgroup$
    – Maria Gerasimova
    33 mins ago










  • $begingroup$
    @MariaGerasimova you have to discard the focal case. Writing, for instance, the solvable group $mathbf{Z}wrmathbf{Z}$ as an ascending HNN extension (w.r.t. an endomorphism with image of finite index), you make it act vertex-transitively on a tree with infinite valency (with abelian stabilizers). So you need to explicitly exclude the case of a fixed point at infinity.
    $endgroup$
    – YCor
    17 mins ago


















$begingroup$
Thank you! And if I assume that stabilizers are amenable, will it be the same?
$endgroup$
– Maria Gerasimova
33 mins ago




$begingroup$
Thank you! And if I assume that stabilizers are amenable, will it be the same?
$endgroup$
– Maria Gerasimova
33 mins ago












$begingroup$
@MariaGerasimova you have to discard the focal case. Writing, for instance, the solvable group $mathbf{Z}wrmathbf{Z}$ as an ascending HNN extension (w.r.t. an endomorphism with image of finite index), you make it act vertex-transitively on a tree with infinite valency (with abelian stabilizers). So you need to explicitly exclude the case of a fixed point at infinity.
$endgroup$
– YCor
17 mins ago






$begingroup$
@MariaGerasimova you have to discard the focal case. Writing, for instance, the solvable group $mathbf{Z}wrmathbf{Z}$ as an ascending HNN extension (w.r.t. an endomorphism with image of finite index), you make it act vertex-transitively on a tree with infinite valency (with abelian stabilizers). So you need to explicitly exclude the case of a fixed point at infinity.
$endgroup$
– YCor
17 mins ago




















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