Groups acting on trees
$begingroup$
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
asked 4 hours ago
Maria GerasimovaMaria Gerasimova
30617
$endgroup$
add a comment |
$begingroup$
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
asked 4 hours ago
Maria GerasimovaMaria Gerasimova
30617
$endgroup$
$begingroup$
Properly implies finite stabilizers
$endgroup$
– YCor
4 hours ago
add a comment |
$begingroup$
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
asked 4 hours ago
Maria GerasimovaMaria Gerasimova
30617
$endgroup$
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
asked 4 hours ago
Maria GerasimovaMaria Gerasimova
30617
asked 4 hours ago
Maria GerasimovaMaria Gerasimova
30617
asked 4 hours ago
Maria GerasimovaMaria Gerasimova
30617
asked 4 hours ago
Maria GerasimovaMaria Gerasimova
30617
asked 4 hours ago
Maria GerasimovaMaria Gerasimova
30617
30617
$begingroup$
Properly implies finite stabilizers
$endgroup$
– YCor
4 hours ago
add a comment |
$begingroup$
Properly implies finite stabilizers
$endgroup$
– YCor
4 hours ago
$begingroup$
Properly implies finite stabilizers
$endgroup$
– YCor
4 hours ago
$begingroup$
Properly implies finite stabilizers
$endgroup$
– YCor
4 hours ago
add a comment |
1 Answer
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$begingroup$
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.
answered 4 hours ago
YCorYCor
27.9k482134
$endgroup$
$begingroup$
Thank you! And if I assume that stabilizers are amenable, will it be the same?
$endgroup$
– Maria Gerasimova
4 hours 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
3 hours ago
add a comment |
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1 Answer
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1 Answer
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$begingroup$
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.
answered 4 hours ago
YCorYCor
27.9k482134
$endgroup$
$begingroup$
Thank you! And if I assume that stabilizers are amenable, will it be the same?
$endgroup$
– Maria Gerasimova
4 hours 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
3 hours ago
add a comment |
$begingroup$
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.
answered 4 hours ago
YCorYCor
27.9k482134
$endgroup$
$begingroup$
Thank you! And if I assume that stabilizers are amenable, will it be the same?
$endgroup$
– Maria Gerasimova
4 hours 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
3 hours ago
add a comment |
$begingroup$
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.
answered 4 hours ago
YCorYCor
27.9k482134
$endgroup$
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.
answered 4 hours ago
YCorYCor
27.9k482134
answered 4 hours ago
YCorYCor
27.9k482134
answered 4 hours ago
YCorYCor
27.9k482134
answered 4 hours ago
YCorYCor
27.9k482134
27.9k482134
$begingroup$
Thank you! And if I assume that stabilizers are amenable, will it be the same?
$endgroup$
– Maria Gerasimova
4 hours 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
3 hours ago
add a comment |
$begingroup$
Thank you! And if I assume that stabilizers are amenable, will it be the same?
$endgroup$
– Maria Gerasimova
4 hours 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
3 hours ago
$begingroup$
Thank you! And if I assume that stabilizers are amenable, will it be the same?
$endgroup$
– Maria Gerasimova
4 hours ago
$begingroup$
Thank you! And if I assume that stabilizers are amenable, will it be the same?
$endgroup$
– Maria Gerasimova
4 hours 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
3 hours 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
3 hours ago
add a comment |
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$begingroup$
Properly implies finite stabilizers
$endgroup$
– YCor
4 hours ago