Groups acting on treesSubgroups of p-groupsTrees in groups of exponential growthDo all possible trees arise...



Groups acting on trees


Subgroups of p-groupsTrees in groups of exponential growthDo all possible trees arise as orbit trees of some permutation groups?Subgroups of amenable periodic groups Is there a better description of this class of discrete groups?Groups acting on non-locally-finite trees with independence and specified local actionsWhich finite simple groups can be characterized by their action on a small set?On $ p $-groups with at least one element of order $ p^{2} $Group acting freely on treeFinite groups containing no subgroups of a given order or index













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?










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$endgroup$












  • $begingroup$
    Properly implies finite stabilizers
    $endgroup$
    – YCor
    2 hours 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?










share|cite|improve this question









$endgroup$












  • $begingroup$
    Properly implies finite stabilizers
    $endgroup$
    – YCor
    2 hours 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?










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






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share|cite|improve this question




share|cite|improve this question










asked 2 hours ago









Maria GerasimovaMaria Gerasimova

1967




1967












  • $begingroup$
    Properly implies finite stabilizers
    $endgroup$
    – YCor
    2 hours ago


















  • $begingroup$
    Properly implies finite stabilizers
    $endgroup$
    – YCor
    2 hours ago
















$begingroup$
Properly implies finite stabilizers
$endgroup$
– YCor
2 hours ago




$begingroup$
Properly implies finite stabilizers
$endgroup$
– YCor
2 hours 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
    1 hour 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
    1 hour 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
    1 hour 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
    1 hour 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
    1 hour 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
    1 hour 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 2 hours 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
    1 hour 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
    1 hour ago




















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


















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




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




















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