ashot minasyan (joint work with yago...
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![Page 1: Ashot Minasyan (Joint work with Yago Antolín)reh.math.uni-duesseldorf.de/~gcgta/slides/Minasyan.pdf · (Joint work with Yago Antolín) University of Southampton Düsseldorf, 30.07.2012](https://reader034.vdocuments.us/reader034/viewer/2022050523/5fa7337a567d5a57e929d96c/html5/thumbnails/1.jpg)
Tits alternatives for graph products
Ashot Minasyan(Joint work with Yago Antolín)
University of Southampton
Düsseldorf, 30.07.2012
Ashot Minasyan Tits alternatives for graph products
![Page 2: Ashot Minasyan (Joint work with Yago Antolín)reh.math.uni-duesseldorf.de/~gcgta/slides/Minasyan.pdf · (Joint work with Yago Antolín) University of Southampton Düsseldorf, 30.07.2012](https://reader034.vdocuments.us/reader034/viewer/2022050523/5fa7337a567d5a57e929d96c/html5/thumbnails/2.jpg)
Background and motivation
Theorem (J. Tits, 1972)
Let H be a finitely generated subgroup of GLn(F ) for some field F .Then either H is virtually solvable or H contains a non-abelian freesubgroup.
Ashot Minasyan Tits alternatives for graph products
![Page 3: Ashot Minasyan (Joint work with Yago Antolín)reh.math.uni-duesseldorf.de/~gcgta/slides/Minasyan.pdf · (Joint work with Yago Antolín) University of Southampton Düsseldorf, 30.07.2012](https://reader034.vdocuments.us/reader034/viewer/2022050523/5fa7337a567d5a57e929d96c/html5/thumbnails/3.jpg)
Background and motivation
Theorem (J. Tits, 1972)
Let H be a finitely generated subgroup of GLn(F ) for some field F .Then either H is virtually solvable or H contains a non-abelian freesubgroup.
Ashot Minasyan Tits alternatives for graph products
![Page 4: Ashot Minasyan (Joint work with Yago Antolín)reh.math.uni-duesseldorf.de/~gcgta/slides/Minasyan.pdf · (Joint work with Yago Antolín) University of Southampton Düsseldorf, 30.07.2012](https://reader034.vdocuments.us/reader034/viewer/2022050523/5fa7337a567d5a57e929d96c/html5/thumbnails/4.jpg)
Background and motivation
Theorem (J. Tits, 1972)
Let H be a finitely generated subgroup of GLn(F ) for some field F .Then either H is virtually solvable or H contains a non-abelian freesubgroup.
Ashot Minasyan Tits alternatives for graph products
![Page 5: Ashot Minasyan (Joint work with Yago Antolín)reh.math.uni-duesseldorf.de/~gcgta/slides/Minasyan.pdf · (Joint work with Yago Antolín) University of Southampton Düsseldorf, 30.07.2012](https://reader034.vdocuments.us/reader034/viewer/2022050523/5fa7337a567d5a57e929d96c/html5/thumbnails/5.jpg)
Background and motivation
Theorem (J. Tits, 1972)
Let H be a finitely generated subgroup of GLn(F ) for some field F .Then either H is virtually solvable or H contains a non-abelian freesubgroup.
Ashot Minasyan Tits alternatives for graph products
![Page 6: Ashot Minasyan (Joint work with Yago Antolín)reh.math.uni-duesseldorf.de/~gcgta/slides/Minasyan.pdf · (Joint work with Yago Antolín) University of Southampton Düsseldorf, 30.07.2012](https://reader034.vdocuments.us/reader034/viewer/2022050523/5fa7337a567d5a57e929d96c/html5/thumbnails/6.jpg)
Background and motivation
Theorem (J. Tits, 1972)
Let H be a finitely generated subgroup of GLn(F ) for some field F .Then either H is virtually solvable or H contains a non-abelian freesubgroup.
Similar results have later been proved for other classes of groups. Wewere motivated by
Ashot Minasyan Tits alternatives for graph products
![Page 7: Ashot Minasyan (Joint work with Yago Antolín)reh.math.uni-duesseldorf.de/~gcgta/slides/Minasyan.pdf · (Joint work with Yago Antolín) University of Southampton Düsseldorf, 30.07.2012](https://reader034.vdocuments.us/reader034/viewer/2022050523/5fa7337a567d5a57e929d96c/html5/thumbnails/7.jpg)
Background and motivation
Theorem (J. Tits, 1972)
Let H be a finitely generated subgroup of GLn(F ) for some field F .Then either H is virtually solvable or H contains a non-abelian freesubgroup.
Similar results have later been proved for other classes of groups. Wewere motivated by
Theorem (Noskov-Vinberg, 2002)
Every subgroup of a finitely generated Coxeter group is eithervirtually abelian or large.
Ashot Minasyan Tits alternatives for graph products
![Page 8: Ashot Minasyan (Joint work with Yago Antolín)reh.math.uni-duesseldorf.de/~gcgta/slides/Minasyan.pdf · (Joint work with Yago Antolín) University of Southampton Düsseldorf, 30.07.2012](https://reader034.vdocuments.us/reader034/viewer/2022050523/5fa7337a567d5a57e929d96c/html5/thumbnails/8.jpg)
Background and motivation
Theorem (J. Tits, 1972)
Let H be a finitely generated subgroup of GLn(F ) for some field F .Then either H is virtually solvable or H contains a non-abelian freesubgroup.
Similar results have later been proved for other classes of groups. Wewere motivated by
Theorem (Noskov-Vinberg, 2002)
Every subgroup of a finitely generated Coxeter group is eithervirtually abelian or large.
Recall: a group G is large is there is a finite index subgroup K 6 Gs.t. K maps onto F2.
Ashot Minasyan Tits alternatives for graph products
![Page 9: Ashot Minasyan (Joint work with Yago Antolín)reh.math.uni-duesseldorf.de/~gcgta/slides/Minasyan.pdf · (Joint work with Yago Antolín) University of Southampton Düsseldorf, 30.07.2012](https://reader034.vdocuments.us/reader034/viewer/2022050523/5fa7337a567d5a57e929d96c/html5/thumbnails/9.jpg)
Various forms of Tits Alternative
Definition
Let C be a class of gps. A gp. G satisfies the Tits Alternative rel. to Cif for any f.g. sbgp. H 6 G either H ∈ C or H contains a copy of F2.
Ashot Minasyan Tits alternatives for graph products
![Page 10: Ashot Minasyan (Joint work with Yago Antolín)reh.math.uni-duesseldorf.de/~gcgta/slides/Minasyan.pdf · (Joint work with Yago Antolín) University of Southampton Düsseldorf, 30.07.2012](https://reader034.vdocuments.us/reader034/viewer/2022050523/5fa7337a567d5a57e929d96c/html5/thumbnails/10.jpg)
Various forms of Tits Alternative
Definition
Let C be a class of gps. A gp. G satisfies the Tits Alternative rel. to Cif for any f.g. sbgp. H 6 G either H ∈ C or H contains a copy of F2.
Thus Tits’s result tells us that GLn(F ) satisfies the Tits Alternative rel.to Cvsol .
Ashot Minasyan Tits alternatives for graph products
![Page 11: Ashot Minasyan (Joint work with Yago Antolín)reh.math.uni-duesseldorf.de/~gcgta/slides/Minasyan.pdf · (Joint work with Yago Antolín) University of Southampton Düsseldorf, 30.07.2012](https://reader034.vdocuments.us/reader034/viewer/2022050523/5fa7337a567d5a57e929d96c/html5/thumbnails/11.jpg)
Various forms of Tits Alternative
Definition
Let C be a class of gps. A gp. G satisfies the Tits Alternative rel. to Cif for any f.g. sbgp. H 6 G either H ∈ C or H contains a copy of F2.
Thus Tits’s result tells us that GLn(F ) satisfies the Tits Alternative rel.to Cvsol .
Definition
Let C be a class of gps. A gp. G satisfies the Strong Tits Alternativerel. to C if for any f.g. sbgp. H 6 G either H ∈ C or H is large.
Ashot Minasyan Tits alternatives for graph products
![Page 12: Ashot Minasyan (Joint work with Yago Antolín)reh.math.uni-duesseldorf.de/~gcgta/slides/Minasyan.pdf · (Joint work with Yago Antolín) University of Southampton Düsseldorf, 30.07.2012](https://reader034.vdocuments.us/reader034/viewer/2022050523/5fa7337a567d5a57e929d96c/html5/thumbnails/12.jpg)
Various forms of Tits Alternative
Definition
Let C be a class of gps. A gp. G satisfies the Tits Alternative rel. to Cif for any f.g. sbgp. H 6 G either H ∈ C or H contains a copy of F2.
Thus Tits’s result tells us that GLn(F ) satisfies the Tits Alternative rel.to Cvsol .
Definition
Let C be a class of gps. A gp. G satisfies the Strong Tits Alternativerel. to C if for any f.g. sbgp. H 6 G either H ∈ C or H is large.
The thm. of Noskov-Vinberg claims that Coxeter gps. satisfy theStrong Tits Alternative rel. to Cvab.
Ashot Minasyan Tits alternatives for graph products
![Page 13: Ashot Minasyan (Joint work with Yago Antolín)reh.math.uni-duesseldorf.de/~gcgta/slides/Minasyan.pdf · (Joint work with Yago Antolín) University of Southampton Düsseldorf, 30.07.2012](https://reader034.vdocuments.us/reader034/viewer/2022050523/5fa7337a567d5a57e929d96c/html5/thumbnails/13.jpg)
Graph products of groups
Graph products naturally generalize free and direct products.
Ashot Minasyan Tits alternatives for graph products
![Page 14: Ashot Minasyan (Joint work with Yago Antolín)reh.math.uni-duesseldorf.de/~gcgta/slides/Minasyan.pdf · (Joint work with Yago Antolín) University of Southampton Düsseldorf, 30.07.2012](https://reader034.vdocuments.us/reader034/viewer/2022050523/5fa7337a567d5a57e929d96c/html5/thumbnails/14.jpg)
Graph products of groups
Graph products naturally generalize free and direct products.
Let Γ be a graph and let G = {Gv | v ∈ VΓ} be a family of gps.
Ashot Minasyan Tits alternatives for graph products
![Page 15: Ashot Minasyan (Joint work with Yago Antolín)reh.math.uni-duesseldorf.de/~gcgta/slides/Minasyan.pdf · (Joint work with Yago Antolín) University of Southampton Düsseldorf, 30.07.2012](https://reader034.vdocuments.us/reader034/viewer/2022050523/5fa7337a567d5a57e929d96c/html5/thumbnails/15.jpg)
Graph products of groups
Graph products naturally generalize free and direct products.
Let Γ be a graph and let G = {Gv | v ∈ VΓ} be a family of gps.
The graph product ΓG is obtained from the free product ∗v∈VΓGv byadding the relations
[a, b] = 1 ∀a ∈ Gu, ∀b ∈ Gv whenever (u, v) ∈ EΓ.
Ashot Minasyan Tits alternatives for graph products
![Page 16: Ashot Minasyan (Joint work with Yago Antolín)reh.math.uni-duesseldorf.de/~gcgta/slides/Minasyan.pdf · (Joint work with Yago Antolín) University of Southampton Düsseldorf, 30.07.2012](https://reader034.vdocuments.us/reader034/viewer/2022050523/5fa7337a567d5a57e929d96c/html5/thumbnails/16.jpg)
Graph products of groups
Graph products naturally generalize free and direct products.
Let Γ be a graph and let G = {Gv | v ∈ VΓ} be a family of gps.
The graph product ΓG is obtained from the free product ∗v∈VΓGv byadding the relations
[a, b] = 1 ∀a ∈ Gu, ∀b ∈ Gv whenever (u, v) ∈ EΓ.
Basic examples of graph products are
Ashot Minasyan Tits alternatives for graph products
![Page 17: Ashot Minasyan (Joint work with Yago Antolín)reh.math.uni-duesseldorf.de/~gcgta/slides/Minasyan.pdf · (Joint work with Yago Antolín) University of Southampton Düsseldorf, 30.07.2012](https://reader034.vdocuments.us/reader034/viewer/2022050523/5fa7337a567d5a57e929d96c/html5/thumbnails/17.jpg)
Graph products of groups
Graph products naturally generalize free and direct products.
Let Γ be a graph and let G = {Gv | v ∈ VΓ} be a family of gps.
The graph product ΓG is obtained from the free product ∗v∈VΓGv byadding the relations
[a, b] = 1 ∀a ∈ Gu, ∀b ∈ Gv whenever (u, v) ∈ EΓ.
Basic examples of graph products are
right angled Artin gps. [RAAGs]
Ashot Minasyan Tits alternatives for graph products
![Page 18: Ashot Minasyan (Joint work with Yago Antolín)reh.math.uni-duesseldorf.de/~gcgta/slides/Minasyan.pdf · (Joint work with Yago Antolín) University of Southampton Düsseldorf, 30.07.2012](https://reader034.vdocuments.us/reader034/viewer/2022050523/5fa7337a567d5a57e929d96c/html5/thumbnails/18.jpg)
Graph products of groups
Graph products naturally generalize free and direct products.
Let Γ be a graph and let G = {Gv | v ∈ VΓ} be a family of gps.
The graph product ΓG is obtained from the free product ∗v∈VΓGv byadding the relations
[a, b] = 1 ∀a ∈ Gu, ∀b ∈ Gv whenever (u, v) ∈ EΓ.
Basic examples of graph products are
right angled Artin gps. [RAAGs], if all vertex gps. are Z;
Ashot Minasyan Tits alternatives for graph products
![Page 19: Ashot Minasyan (Joint work with Yago Antolín)reh.math.uni-duesseldorf.de/~gcgta/slides/Minasyan.pdf · (Joint work with Yago Antolín) University of Southampton Düsseldorf, 30.07.2012](https://reader034.vdocuments.us/reader034/viewer/2022050523/5fa7337a567d5a57e929d96c/html5/thumbnails/19.jpg)
Graph products of groups
Graph products naturally generalize free and direct products.
Let Γ be a graph and let G = {Gv | v ∈ VΓ} be a family of gps.
The graph product ΓG is obtained from the free product ∗v∈VΓGv byadding the relations
[a, b] = 1 ∀a ∈ Gu, ∀b ∈ Gv whenever (u, v) ∈ EΓ.
Basic examples of graph products are
right angled Artin gps. [RAAGs], if all vertex gps. are Z;
right angled Coxeter gps.
Ashot Minasyan Tits alternatives for graph products
![Page 20: Ashot Minasyan (Joint work with Yago Antolín)reh.math.uni-duesseldorf.de/~gcgta/slides/Minasyan.pdf · (Joint work with Yago Antolín) University of Southampton Düsseldorf, 30.07.2012](https://reader034.vdocuments.us/reader034/viewer/2022050523/5fa7337a567d5a57e929d96c/html5/thumbnails/20.jpg)
Graph products of groups
Graph products naturally generalize free and direct products.
Let Γ be a graph and let G = {Gv | v ∈ VΓ} be a family of gps.
The graph product ΓG is obtained from the free product ∗v∈VΓGv byadding the relations
[a, b] = 1 ∀a ∈ Gu, ∀b ∈ Gv whenever (u, v) ∈ EΓ.
Basic examples of graph products are
right angled Artin gps. [RAAGs], if all vertex gps. are Z;
right angled Coxeter gps., if all vertex gps. are Z/2Z;
Ashot Minasyan Tits alternatives for graph products
![Page 21: Ashot Minasyan (Joint work with Yago Antolín)reh.math.uni-duesseldorf.de/~gcgta/slides/Minasyan.pdf · (Joint work with Yago Antolín) University of Southampton Düsseldorf, 30.07.2012](https://reader034.vdocuments.us/reader034/viewer/2022050523/5fa7337a567d5a57e929d96c/html5/thumbnails/21.jpg)
Graph products of groups
Graph products naturally generalize free and direct products.
Let Γ be a graph and let G = {Gv | v ∈ VΓ} be a family of gps.
The graph product ΓG is obtained from the free product ∗v∈VΓGv byadding the relations
[a, b] = 1 ∀a ∈ Gu, ∀b ∈ Gv whenever (u, v) ∈ EΓ.
Basic examples of graph products are
right angled Artin gps. [RAAGs], if all vertex gps. are Z;
right angled Coxeter gps., if all vertex gps. are Z/2Z;
If A ⊆ VΓ and ΓA is the full subgraph of Γ spanned by A thenGA := {Gv | v ∈ A} generates a special subgroup GA of G = ΓGwhich is naturally isomorphic to ΓAGA.
Ashot Minasyan Tits alternatives for graph products
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Tits Alternative for graph products
Consider the following properties of the class of groups C:
Ashot Minasyan Tits alternatives for graph products
![Page 23: Ashot Minasyan (Joint work with Yago Antolín)reh.math.uni-duesseldorf.de/~gcgta/slides/Minasyan.pdf · (Joint work with Yago Antolín) University of Southampton Düsseldorf, 30.07.2012](https://reader034.vdocuments.us/reader034/viewer/2022050523/5fa7337a567d5a57e929d96c/html5/thumbnails/23.jpg)
Tits Alternative for graph products
Consider the following properties of the class of groups C:
(P0) if L ∈ C and M ∼= L then M ∈ C;
Ashot Minasyan Tits alternatives for graph products
![Page 24: Ashot Minasyan (Joint work with Yago Antolín)reh.math.uni-duesseldorf.de/~gcgta/slides/Minasyan.pdf · (Joint work with Yago Antolín) University of Southampton Düsseldorf, 30.07.2012](https://reader034.vdocuments.us/reader034/viewer/2022050523/5fa7337a567d5a57e929d96c/html5/thumbnails/24.jpg)
Tits Alternative for graph products
Consider the following properties of the class of groups C:
(P0) if L ∈ C and M ∼= L then M ∈ C;
(P1) if L ∈ C and M 6 L is f.g. then M ∈ C;
Ashot Minasyan Tits alternatives for graph products
![Page 25: Ashot Minasyan (Joint work with Yago Antolín)reh.math.uni-duesseldorf.de/~gcgta/slides/Minasyan.pdf · (Joint work with Yago Antolín) University of Southampton Düsseldorf, 30.07.2012](https://reader034.vdocuments.us/reader034/viewer/2022050523/5fa7337a567d5a57e929d96c/html5/thumbnails/25.jpg)
Tits Alternative for graph products
Consider the following properties of the class of groups C:
(P0) if L ∈ C and M ∼= L then M ∈ C;
(P1) if L ∈ C and M 6 L is f.g. then M ∈ C;
(P2) if L,M ∈ C are f.g. then L × M ∈ C;
Ashot Minasyan Tits alternatives for graph products
![Page 26: Ashot Minasyan (Joint work with Yago Antolín)reh.math.uni-duesseldorf.de/~gcgta/slides/Minasyan.pdf · (Joint work with Yago Antolín) University of Southampton Düsseldorf, 30.07.2012](https://reader034.vdocuments.us/reader034/viewer/2022050523/5fa7337a567d5a57e929d96c/html5/thumbnails/26.jpg)
Tits Alternative for graph products
Consider the following properties of the class of groups C:
(P0) if L ∈ C and M ∼= L then M ∈ C;
(P1) if L ∈ C and M 6 L is f.g. then M ∈ C;
(P2) if L,M ∈ C are f.g. then L × M ∈ C;
(P3) Z ∈ C;
Ashot Minasyan Tits alternatives for graph products
![Page 27: Ashot Minasyan (Joint work with Yago Antolín)reh.math.uni-duesseldorf.de/~gcgta/slides/Minasyan.pdf · (Joint work with Yago Antolín) University of Southampton Düsseldorf, 30.07.2012](https://reader034.vdocuments.us/reader034/viewer/2022050523/5fa7337a567d5a57e929d96c/html5/thumbnails/27.jpg)
Tits Alternative for graph products
Consider the following properties of the class of groups C:
(P0) if L ∈ C and M ∼= L then M ∈ C;
(P1) if L ∈ C and M 6 L is f.g. then M ∈ C;
(P2) if L,M ∈ C are f.g. then L × M ∈ C;
(P3) Z ∈ C;
(P4) if Z/2Z ∈ C then D∞ ∈ C.
Ashot Minasyan Tits alternatives for graph products
![Page 28: Ashot Minasyan (Joint work with Yago Antolín)reh.math.uni-duesseldorf.de/~gcgta/slides/Minasyan.pdf · (Joint work with Yago Antolín) University of Southampton Düsseldorf, 30.07.2012](https://reader034.vdocuments.us/reader034/viewer/2022050523/5fa7337a567d5a57e929d96c/html5/thumbnails/28.jpg)
Tits Alternative for graph products
Consider the following properties of the class of groups C:
(P0) if L ∈ C and M ∼= L then M ∈ C;
(P1) if L ∈ C and M 6 L is f.g. then M ∈ C;
(P2) if L,M ∈ C are f.g. then L × M ∈ C;
(P3) Z ∈ C;
(P4) if Z/2Z ∈ C then D∞ ∈ C.
Theorem A (Antolín-M.)
Let C be a class of gps. with (P0)–(P4). Then a graph productG = ΓG satisfies the Tits Alternative rel. to C iff each Gv , v ∈ VΓ,satisfies this alternative.
Ashot Minasyan Tits alternatives for graph products
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Tits Alternative for graph products
Consider the following properties of the class of groups C:
(P0) if L ∈ C and M ∼= L then M ∈ C;
(P1) if L ∈ C and M 6 L is f.g. then M ∈ C;
(P2) if L,M ∈ C are f.g. then L × M ∈ C;
(P3) Z ∈ C;
(P4) if Z/2Z ∈ C then D∞ ∈ C.
Theorem A (Antolín-M.)
Let C be a class of gps. with (P0)–(P4). Then a graph productG = ΓG satisfies the Tits Alternative rel. to C iff each Gv , v ∈ VΓ,satisfies this alternative.
Evidently the conditions (P0)–(P4) are necessary.
Ashot Minasyan Tits alternatives for graph products
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Tits Alternative for graph products
Consider the following properties of the class of groups C:
(P0) if L ∈ C and M ∼= L then M ∈ C;
(P1) if L ∈ C and M 6 L is f.g. then M ∈ C;
(P2) if L,M ∈ C are f.g. then L × M ∈ C;
(P3) Z ∈ C;
(P4) if Z/2Z ∈ C then D∞ ∈ C.
Theorem A (Antolín-M.)
Let C be a class of gps. with (P0)–(P4). Then a graph productG = ΓG satisfies the Tits Alternative rel. to C iff each Gv , v ∈ VΓ,satisfies this alternative.
Corollary
If all vertex gps. are linear then G = ΓG satisfies the Tits Alternativerel. to Cvsol .
Ashot Minasyan Tits alternatives for graph products
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Strong Tits Alternative for graph products
(P0) if L ∈ C and M ∼= L then M ∈ C;
(P1) if L ∈ C and M 6 L is f.g. then M ∈ C;
(P2) if L,M ∈ C are f.g. then L × M ∈ C;
(P3) Z ∈ C;
(P4) if Z/2Z ∈ C then D∞ ∈ C;
Ashot Minasyan Tits alternatives for graph products
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Strong Tits Alternative for graph products
(P0) if L ∈ C and M ∼= L then M ∈ C;
(P1) if L ∈ C and M 6 L is f.g. then M ∈ C;
(P2) if L,M ∈ C are f.g. then L × M ∈ C;
(P3) Z ∈ C;
(P4) if Z/2Z ∈ C then D∞ ∈ C;
(P5) if L ∈ C is non-trivial and f.g. then L contains a proper f.i. sbgp.
Ashot Minasyan Tits alternatives for graph products
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Strong Tits Alternative for graph products
(P0) if L ∈ C and M ∼= L then M ∈ C;
(P1) if L ∈ C and M 6 L is f.g. then M ∈ C;
(P2) if L,M ∈ C are f.g. then L × M ∈ C;
(P3) Z ∈ C;
(P4) if Z/2Z ∈ C then D∞ ∈ C;
(P5) if L ∈ C is non-trivial and f.g. then L contains a proper f.i. sbgp.
Theorem B (Antolín-M.)
Let C be a class of gps. with (P0)–(P5). Then a graph productG = ΓG satisfies the Strong Tits Alternative rel. to C iff each Gv ,v ∈ VΓ, satisfies this alternative.
Ashot Minasyan Tits alternatives for graph products
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Strong Tits Alternative for graph products
(P0) if L ∈ C and M ∼= L then M ∈ C;
(P1) if L ∈ C and M 6 L is f.g. then M ∈ C;
(P2) if L,M ∈ C are f.g. then L × M ∈ C;
(P3) Z ∈ C;
(P4) if Z/2Z ∈ C then D∞ ∈ C;
(P5) if L ∈ C is non-trivial and f.g. then L contains a proper f.i. sbgp.
Theorem B (Antolín-M.)
Let C be a class of gps. with (P0)–(P5). Then a graph productG = ΓG satisfies the Strong Tits Alternative rel. to C iff each Gv ,v ∈ VΓ, satisfies this alternative.
(P5) is necessary, b/c if L 6= {1} has no proper f.i. sbgps., then L ∗ Lcannot be large.
Ashot Minasyan Tits alternatives for graph products
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Strong Tits Alternative for graph products
(P0) if L ∈ C and M ∼= L then M ∈ C;
(P1) if L ∈ C and M 6 L is f.g. then M ∈ C;
(P2) if L,M ∈ C are f.g. then L × M ∈ C;
(P3) Z ∈ C;
(P4) if Z/2Z ∈ C then D∞ ∈ C;
(P5) if L ∈ C is non-trivial and f.g. then L contains a proper f.i. sbgp.
Theorem B (Antolín-M.)
Let C be a class of gps. with (P0)–(P5). Then a graph productG = ΓG satisfies the Strong Tits Alternative rel. to C iff each Gv ,v ∈ VΓ, satisfies this alternative.
Examples of gps. with (P0)–(P5):
Ashot Minasyan Tits alternatives for graph products
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Strong Tits Alternative for graph products
(P0) if L ∈ C and M ∼= L then M ∈ C;
(P1) if L ∈ C and M 6 L is f.g. then M ∈ C;
(P2) if L,M ∈ C are f.g. then L × M ∈ C;
(P3) Z ∈ C;
(P4) if Z/2Z ∈ C then D∞ ∈ C;
(P5) if L ∈ C is non-trivial and f.g. then L contains a proper f.i. sbgp.
Theorem B (Antolín-M.)
Let C be a class of gps. with (P0)–(P5). Then a graph productG = ΓG satisfies the Strong Tits Alternative rel. to C iff each Gv ,v ∈ VΓ, satisfies this alternative.
Examples of gps. with (P0)–(P5): virt. abelian gps,
Ashot Minasyan Tits alternatives for graph products
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Strong Tits Alternative for graph products
(P0) if L ∈ C and M ∼= L then M ∈ C;
(P1) if L ∈ C and M 6 L is f.g. then M ∈ C;
(P2) if L,M ∈ C are f.g. then L × M ∈ C;
(P3) Z ∈ C;
(P4) if Z/2Z ∈ C then D∞ ∈ C;
(P5) if L ∈ C is non-trivial and f.g. then L contains a proper f.i. sbgp.
Theorem B (Antolín-M.)
Let C be a class of gps. with (P0)–(P5). Then a graph productG = ΓG satisfies the Strong Tits Alternative rel. to C iff each Gv ,v ∈ VΓ, satisfies this alternative.
Examples of gps. with (P0)–(P5): virt. abelian gps, (virt.) polycyclicgps.,
Ashot Minasyan Tits alternatives for graph products
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Strong Tits Alternative for graph products
(P0) if L ∈ C and M ∼= L then M ∈ C;
(P1) if L ∈ C and M 6 L is f.g. then M ∈ C;
(P2) if L,M ∈ C are f.g. then L × M ∈ C;
(P3) Z ∈ C;
(P4) if Z/2Z ∈ C then D∞ ∈ C;
(P5) if L ∈ C is non-trivial and f.g. then L contains a proper f.i. sbgp.
Theorem B (Antolín-M.)
Let C be a class of gps. with (P0)–(P5). Then a graph productG = ΓG satisfies the Strong Tits Alternative rel. to C iff each Gv ,v ∈ VΓ, satisfies this alternative.
Examples of gps. with (P0)–(P5): virt. abelian gps, (virt.) polycyclicgps., virt. nilpotent gps.,
Ashot Minasyan Tits alternatives for graph products
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Strong Tits Alternative for graph products
(P0) if L ∈ C and M ∼= L then M ∈ C;
(P1) if L ∈ C and M 6 L is f.g. then M ∈ C;
(P2) if L,M ∈ C are f.g. then L × M ∈ C;
(P3) Z ∈ C;
(P4) if Z/2Z ∈ C then D∞ ∈ C;
(P5) if L ∈ C is non-trivial and f.g. then L contains a proper f.i. sbgp.
Theorem B (Antolín-M.)
Let C be a class of gps. with (P0)–(P5). Then a graph productG = ΓG satisfies the Strong Tits Alternative rel. to C iff each Gv ,v ∈ VΓ, satisfies this alternative.
Examples of gps. with (P0)–(P5): virt. abelian gps, (virt.) polycyclicgps., virt. nilpotent gps., (virt.) solvable gps.,
Ashot Minasyan Tits alternatives for graph products
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Strong Tits Alternative for graph products
(P0) if L ∈ C and M ∼= L then M ∈ C;
(P1) if L ∈ C and M 6 L is f.g. then M ∈ C;
(P2) if L,M ∈ C are f.g. then L × M ∈ C;
(P3) Z ∈ C;
(P4) if Z/2Z ∈ C then D∞ ∈ C;
(P5) if L ∈ C is non-trivial and f.g. then L contains a proper f.i. sbgp.
Theorem B (Antolín-M.)
Let C be a class of gps. with (P0)–(P5). Then a graph productG = ΓG satisfies the Strong Tits Alternative rel. to C iff each Gv ,v ∈ VΓ, satisfies this alternative.
Examples of gps. with (P0)–(P5): virt. abelian gps, (virt.) polycyclicgps., virt. nilpotent gps., (virt.) solvable gps., elementary amenablegps.
Ashot Minasyan Tits alternatives for graph products
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Strong Tits Alternative for graph products
(P0) if L ∈ C and M ∼= L then M ∈ C;
(P1) if L ∈ C and M 6 L is f.g. then M ∈ C;
(P2) if L,M ∈ C are f.g. then L × M ∈ C;
(P3) Z ∈ C;
(P4) if Z/2Z ∈ C then D∞ ∈ C;
(P5) if L ∈ C is non-trivial and f.g. then L contains a proper f.i. sbgp.
Theorem B (Antolín-M.)
Let C be a class of gps. with (P0)–(P5). Then a graph productG = ΓG satisfies the Strong Tits Alternative rel. to C iff each Gv ,v ∈ VΓ, satisfies this alternative.
Corollary
Suppose C = Csol−m for some m ≥ 2 or C = Cvsol−n for some n ≥ 1.Let G be a graph product of gps. from C. Then any f.g. sbgp. of Geither belongs to C or is large.
Ashot Minasyan Tits alternatives for graph products
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The Strongest Tits Alternative
Definition
Let C be a class of gps. A gp. G satisfies the Strongest TitsAlternative rel. to C if for any f.g. sbgp. H 6 G either H ∈ C or H mapsonto F2.
Ashot Minasyan Tits alternatives for graph products
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The Strongest Tits Alternative
Definition
Let C be a class of gps. A gp. G satisfies the Strongest TitsAlternative rel. to C if for any f.g. sbgp. H 6 G either H ∈ C or H mapsonto F2.
Example
The gp. G := 〈a, b, c | a2b2 = c2〉 is t.-f. and large but does not maponto F2.
Ashot Minasyan Tits alternatives for graph products
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The Strongest Tits Alternative
Definition
Let C be a class of gps. A gp. G satisfies the Strongest TitsAlternative rel. to C if for any f.g. sbgp. H 6 G either H ∈ C or H mapsonto F2.
Example
The gp. G := 〈a, b, c | a2b2 = c2〉 is t.-f. and large but does not maponto F2.
Example
Any residually free gp. satisfies the Strongest Tits Alternative rel. toCab.
Ashot Minasyan Tits alternatives for graph products
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The Strongest Tits Alternative
Definition
Let C be a class of gps. A gp. G satisfies the Strongest TitsAlternative rel. to C if for any f.g. sbgp. H 6 G either H ∈ C or H mapsonto F2.
Example
The gp. G := 〈a, b, c | a2b2 = c2〉 is t.-f. and large but does not maponto F2.
Example
Any residually free gp. satisfies the Strongest Tits Alternative rel. toCab.
Observe that if L ∗ L maps onto F2 then L must have an epimorphismonto Z.
Ashot Minasyan Tits alternatives for graph products
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Strongest Tits Alternative for graph products
(P0) if L ∈ C and M ∼= L then M ∈ C;
(P1) if L ∈ C and M 6 L is f.g. then M ∈ C;
(P2) if L,M ∈ C are f.g. then L × M ∈ C;
(P3) Z ∈ C;
Ashot Minasyan Tits alternatives for graph products
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Strongest Tits Alternative for graph products
(P0) if L ∈ C and M ∼= L then M ∈ C;
(P1) if L ∈ C and M 6 L is f.g. then M ∈ C;
(P2) if L,M ∈ C are f.g. then L × M ∈ C;
(P3) Z ∈ C;
(P6) if L ∈ C is non-trivial and f.g. then L maps onto Z.
Ashot Minasyan Tits alternatives for graph products
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Strongest Tits Alternative for graph products
(P0) if L ∈ C and M ∼= L then M ∈ C;
(P1) if L ∈ C and M 6 L is f.g. then M ∈ C;
(P2) if L,M ∈ C are f.g. then L × M ∈ C;
(P3) Z ∈ C;
(P6) if L ∈ C is non-trivial and f.g. then L maps onto Z.
Theorem C (Antolín-M.)
Let C be a class of gps. with (P0)–(P3) and (P6). Then a graphproduct G = ΓG satisfies the Strongest Tits Alternative rel. to C iffeach Gv , v ∈ VΓ, satisfies this alternative.
Ashot Minasyan Tits alternatives for graph products
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Strongest Tits Alternative for graph products
(P0) if L ∈ C and M ∼= L then M ∈ C;
(P1) if L ∈ C and M 6 L is f.g. then M ∈ C;
(P2) if L,M ∈ C are f.g. then L × M ∈ C;
(P3) Z ∈ C;
(P6) if L ∈ C is non-trivial and f.g. then L maps onto Z.
Theorem C (Antolín-M.)
Let C be a class of gps. with (P0)–(P3) and (P6). Then a graphproduct G = ΓG satisfies the Strongest Tits Alternative rel. to C iffeach Gv , v ∈ VΓ, satisfies this alternative.
Examples of gps. with (P0)–(P3) and (P6):
Ashot Minasyan Tits alternatives for graph products
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Strongest Tits Alternative for graph products
(P0) if L ∈ C and M ∼= L then M ∈ C;
(P1) if L ∈ C and M 6 L is f.g. then M ∈ C;
(P2) if L,M ∈ C are f.g. then L × M ∈ C;
(P3) Z ∈ C;
(P6) if L ∈ C is non-trivial and f.g. then L maps onto Z.
Theorem C (Antolín-M.)
Let C be a class of gps. with (P0)–(P3) and (P6). Then a graphproduct G = ΓG satisfies the Strongest Tits Alternative rel. to C iffeach Gv , v ∈ VΓ, satisfies this alternative.
Examples of gps. with (P0)–(P3) and (P6): t.-f. abelian gps,
Ashot Minasyan Tits alternatives for graph products
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Strongest Tits Alternative for graph products
(P0) if L ∈ C and M ∼= L then M ∈ C;
(P1) if L ∈ C and M 6 L is f.g. then M ∈ C;
(P2) if L,M ∈ C are f.g. then L × M ∈ C;
(P3) Z ∈ C;
(P6) if L ∈ C is non-trivial and f.g. then L maps onto Z.
Theorem C (Antolín-M.)
Let C be a class of gps. with (P0)–(P3) and (P6). Then a graphproduct G = ΓG satisfies the Strongest Tits Alternative rel. to C iffeach Gv , v ∈ VΓ, satisfies this alternative.
Examples of gps. with (P0)–(P3) and (P6): t.-f. abelian gps, t.-f.nilpotent gps.
Ashot Minasyan Tits alternatives for graph products
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Strongest Tits Alternative for graph products
(P0) if L ∈ C and M ∼= L then M ∈ C;
(P1) if L ∈ C and M 6 L is f.g. then M ∈ C;
(P2) if L,M ∈ C are f.g. then L × M ∈ C;
(P3) Z ∈ C;
(P6) if L ∈ C is non-trivial and f.g. then L maps onto Z.
Theorem C (Antolín-M.)
Let C be a class of gps. with (P0)–(P3) and (P6). Then a graphproduct G = ΓG satisfies the Strongest Tits Alternative rel. to C iffeach Gv , v ∈ VΓ, satisfies this alternative.
Examples of gps. with (P0)–(P3) and (P6): t.-f. abelian gps, t.-f.nilpotent gps.
Corollary
Any f.g. non-abelian sbgp. of a RAAG maps onto F2.
Ashot Minasyan Tits alternatives for graph products
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Strongest Tits Alternative for graph products
(P0) if L ∈ C and M ∼= L then M ∈ C;
(P1) if L ∈ C and M 6 L is f.g. then M ∈ C;
(P2) if L,M ∈ C are f.g. then L × M ∈ C;
(P3) Z ∈ C;
(P6) if L ∈ C is non-trivial and f.g. then L maps onto Z.
Theorem C (Antolín-M.)
Let C be a class of gps. with (P0)–(P3) and (P6). Then a graphproduct G = ΓG satisfies the Strongest Tits Alternative rel. to C iffeach Gv , v ∈ VΓ, satisfies this alternative.
Examples of gps. with (P0)–(P3) and (P6): t.-f. abelian gps, t.-f.nilpotent gps.
Corollary
Any non-abelian sbgp. of a RAAG maps onto F2.
Ashot Minasyan Tits alternatives for graph products
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Applications of Theorem C
Corollary
Any non-abelian sbgp. of a RAAG maps onto F2.
Ashot Minasyan Tits alternatives for graph products
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Applications of Theorem C
Corollary
Any non-abelian sbgp. of a RAAG maps onto F2.
One can use this to recover
Theorem (Baudisch, 1981)
A 2-generator sbgp. of a RAAG is either free or free abelian.
Ashot Minasyan Tits alternatives for graph products
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Applications of Theorem C
Corollary
Any non-abelian sbgp. of a RAAG maps onto F2.
One can use this to recover
Theorem (Baudisch, 1981)
A 2-generator sbgp. of a RAAG is either free or free abelian.
Combining with a result of Lyndon-Schützenberger we also get
Corollary
If G is a RAAG and a, b, c ∈ G satisfy ambn = cp, for m, n, p ≥ 2, thena, b, c pairwise commute.
Ashot Minasyan Tits alternatives for graph products