Noncoherence of lattices
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1 January 20, 2013
2 Definitions and examples Definition A group G is called coherent if every finitely-generated subgroup of G is also finitely-presented.
3 Definitions and examples Definition A group G is called coherent if every finitely-generated subgroup of G is also finitely-presented. Examples of coherent groups: Free groups.
4 Definitions and examples Definition A group G is called coherent if every finitely-generated subgroup of G is also finitely-presented. Examples of coherent groups: Free groups. Surface groups.
5 Definitions and examples Definition A group G is called coherent if every finitely-generated subgroup of G is also finitely-presented. Examples of coherent groups: Free groups. Surface groups. Abelian groups.
6 Definitions and examples Definition A group G is called coherent if every finitely-generated subgroup of G is also finitely-presented. Examples of coherent groups: Free groups. Surface groups. Abelian groups. Polycyclic groups.
7 Definitions and examples Definition A group G is called coherent if every finitely-generated subgroup of G is also finitely-presented. Examples of coherent groups: Free groups. Surface groups. Abelian groups. Polycyclic groups. Fundamental groups of 3-manifolds (Scott; Shalen).
8 Definitions and examples Definition A group G is called coherent if every finitely-generated subgroup of G is also finitely-presented. Examples of coherent groups: Free groups. Surface groups. Abelian groups. Polycyclic groups. Fundamental groups of 3-manifolds (Scott; Shalen). Free-by-cyclic groups (Feighn and Handel).
9 Definitions and examples Definition A group G is called coherent if every finitely-generated subgroup of G is also finitely-presented. Examples of coherent groups: Free groups. Surface groups. Abelian groups. Polycyclic groups. Fundamental groups of 3-manifolds (Scott; Shalen). Free-by-cyclic groups (Feighn and Handel). Certain classes of small cancellation groups (McCammond and Wise). For instance, G = x 1,..., x n W m, where m > W.
10 Nonexamples: Examples of incoherent groups:
11 Nonexamples: Examples of incoherent groups: F 2 F 2. (Stallings-?; Grunewald)
12 Nonexamples: Examples of incoherent groups: F 2 F 2. (Stallings-?; Grunewald) More generally, any right-angled Artin group whose graph is non-chordal, i.e., contains an embedded cycle of length 4 without a cord. (Hermiller and Meier)
13 Nonexamples: Examples of incoherent groups: F 2 F 2. (Stallings-?; Grunewald) More generally, any right-angled Artin group whose graph is non-chordal, i.e., contains an embedded cycle of length 4 without a cord. (Hermiller and Meier) In particular: SL(n, Z), n 4.
14 Nonexamples: Examples of incoherent groups: F 2 F 2. (Stallings-?; Grunewald) More generally, any right-angled Artin group whose graph is non-chordal, i.e., contains an embedded cycle of length 4 without a cord. (Hermiller and Meier) In particular: SL(n, Z), n 4. Rips construction: 2-dimensional hyperbolic groups 1 K G Q 1, where Q is f.p., Q =.
15 Nonexamples: Examples of incoherent groups: F 2 F 2. (Stallings-?; Grunewald) More generally, any right-angled Artin group whose graph is non-chordal, i.e., contains an embedded cycle of length 4 without a cord. (Hermiller and Meier) In particular: SL(n, Z), n 4. Rips construction: 2-dimensional hyperbolic groups 1 K G Q 1, where Q is f.p., Q =. Aut(F 2 ) and the braid group B 4 (Gordon).
16 Nonexamples: Examples of incoherent groups: F 2 F 2. (Stallings-?; Grunewald) More generally, any right-angled Artin group whose graph is non-chordal, i.e., contains an embedded cycle of length 4 without a cord. (Hermiller and Meier) In particular: SL(n, Z), n 4. Rips construction: 2-dimensional hyperbolic groups 1 K G Q 1, where Q is f.p., Q =. Aut(F 2 ) and the braid group B 4 (Gordon). Doubles F H F, where F is free of rank 2 and 2 < F : H <. (Gersten)
17 Nonexamples: Examples of incoherent groups: Problem F 2 F 2. (Stallings-?; Grunewald) More generally, any right-angled Artin group whose graph is non-chordal, i.e., contains an embedded cycle of length 4 without a cord. (Hermiller and Meier) In particular: SL(n, Z), n 4. Rips construction: 2-dimensional hyperbolic groups 1 K G Q 1, where Q is f.p., Q =. Aut(F 2 ) and the braid group B 4 (Gordon). Doubles F H F, where F is free of rank 2 and 2 < F : H <. (Gersten) (Serre, 1977) Are SL(3, Z) and SL(2, Z( 1 p )) noncoherent?
18 General conjecture Definition Let G be a Lie group. A subgroup Γ < G is a lattice if Γ is discrete and Vol(G/Γ) <.
19 General conjecture Definition Let G be a Lie group. A subgroup Γ < G is a lattice if Γ is discrete and Vol(G/Γ) <. Conjecture Let G be a connected semisimple Lie group without compact factors and G not locally isomorphic to SL(2, R) and SL(2, C). Then every lattice Γ in G is non-coherent.
20 General conjecture Definition Let G be a Lie group. A subgroup Γ < G is a lattice if Γ is discrete and Vol(G/Γ) <. Conjecture Let G be a connected semisimple Lie group without compact factors and G not locally isomorphic to SL(2, R) and SL(2, C). Then every lattice Γ in G is non-coherent. Note that lattices in SL(2, R) and SL(2, C) are (virtually) free, surface and 3-manifold groups, so they are coherent.
21 Real-hyperbolic space: G = SO(n, 1)
22 Real-hyperbolic space: G = SO(n, 1) Theorem Let Γ < SO(n, 1) be an arithmetic lattice of the simplest type (associated with a quadratic form over a number field). Then Γ is noncoherent provided that n 4. (Kapovich, Potyagailo, Vinberg; Agol)
23 Real-hyperbolic space: G = SO(n, 1) Theorem Let Γ < SO(n, 1) be an arithmetic lattice of the simplest type (associated with a quadratic form over a number field). Then Γ is noncoherent provided that n 4. (Kapovich, Potyagailo, Vinberg; Agol) Examples: O(x x n 2 2xn+1 2, Z).
24 Real-hyperbolic space: G = SO(n, 1) Theorem Let Γ < SO(n, 1) be an arithmetic lattice of the simplest type (associated with a quadratic form over a number field). Then Γ is noncoherent provided that n 4. (Kapovich, Potyagailo, Vinberg; Agol) Examples: O(x x n 2 2xn+1 2, Z). Theorem Let Γ < SO(n, 1) be an arithmetic lattice of quaternionic type (associated with a hermitian quadratic form over a central 4-dimensional division ring). Then Γ is non-coherent provided that n 4.
25 Corollary Every arithmetic lattice Γ < SO(n, 1) is noncoherent provided that n 4, n 7.
26 Corollary Every arithmetic lattice Γ < SO(n, 1) is noncoherent provided that n 4, n 7. Observation All known constructions of non-arithmetic lattices in SO(n, 1), n 4 (Makarov; Gromov-Piatetsky-Shapiro; Agol) lead to noncoherent groups. (Kapovich, Potyagailo, Vinberg)
27 Complex-hyperbolic space: G = SU(n, 1)
28 Complex-hyperbolic space: G = SU(n, 1) Theorem Let Γ < SU(2, 1) be a cocompact lattice (arithmetic or not) with infinite abelianization. Then Γ is noncoherent.
29 Complex-hyperbolic space: G = SU(n, 1) Theorem Let Γ < SU(2, 1) be a cocompact lattice (arithmetic or not) with infinite abelianization. Then Γ is noncoherent. Corollary Every cocompact arithmetic lattice of the simplest type (associated with a hermitian quadratic form over a number field) in SU(n, 1), n 2, is noncoherent.
30 Complex-hyperbolic space: G = SU(n, 1) Theorem Let Γ < SU(2, 1) be a cocompact lattice (arithmetic or not) with infinite abelianization. Then Γ is noncoherent. Corollary Every cocompact arithmetic lattice of the simplest type (associated with a hermitian quadratic form over a number field) in SU(n, 1), n 2, is noncoherent. Observation All known examples of non-arithmetic lattices in SU(n, 1), n = 2, 3 are noncoherent.
31 Quaternionic and octontionic hyperbolic spaces Theorem Every lattice in Isom(HH n ) and Isom(OH 2 ) is noncoherent. Proof: Reduction to the SO(4, 1), SO(8, 1) cases.
32 Proof in the complex-hyperbolic case Assume Γ is torsion-free. Since H 1 (Γ) 0 and Γ is a Kähler group, 1 Λ Γ Z 2 1.
33 Proof in the complex-hyperbolic case Assume Γ is torsion-free. Since H 1 (Γ) 0 and Γ is a Kähler group, Theorem 1 Λ Γ Z 2 1. (Delzant s Alternative) Either Λ is f.g. or M = CH 2 /Γ holomorphically fibers over a Riemann surface.
34 Proof in the complex-hyperbolic case Assume Γ is torsion-free. Since H 1 (Γ) 0 and Γ is a Kähler group, Theorem 1 Λ Γ Z 2 1. (Delzant s Alternative) Either Λ is f.g. or M = CH 2 /Γ holomorphically fibers over a Riemann surface. If M fibers then the image of the generic fiber group is f.g. but not f.p. (Kapovich).
35 Proof in the complex-hyperbolic case Assume Γ is torsion-free. Since H 1 (Γ) 0 and Γ is a Kähler group, Theorem 1 Λ Γ Z 2 1. (Delzant s Alternative) Either Λ is f.g. or M = CH 2 /Γ holomorphically fibers over a Riemann surface. If M fibers then the image of the generic fiber group is f.g. but not f.p. (Kapovich). If Λ is f.g., then either it is not f.p. (and Γ is noncoherent) or Λ is f.p. and then is a PD(2) group (Hillman).
36 Proof in the complex-hyperbolic case Assume Γ is torsion-free. Since H 1 (Γ) 0 and Γ is a Kähler group, Theorem 1 Λ Γ Z 2 1. (Delzant s Alternative) Either Λ is f.g. or M = CH 2 /Γ holomorphically fibers over a Riemann surface. If M fibers then the image of the generic fiber group is f.g. but not f.p. (Kapovich). If Λ is f.g., then either it is not f.p. (and Γ is noncoherent) or Λ is f.p. and then is a PD(2) group (Hillman). Then Λ is a surface group (Eckmann Linnel Müller).
37 Proof in the complex-hyperbolic case Assume Γ is torsion-free. Since H 1 (Γ) 0 and Γ is a Kähler group, Theorem 1 Λ Γ Z 2 1. (Delzant s Alternative) Either Λ is f.g. or M = CH 2 /Γ holomorphically fibers over a Riemann surface. If M fibers then the image of the generic fiber group is f.g. but not f.p. (Kapovich). If Λ is f.g., then either it is not f.p. (and Γ is noncoherent) or Λ is f.p. and then is a PD(2) group (Hillman). Then Λ is a surface group (Eckmann Linnel Müller). But for any extension of the form 1 π 1 (S) Γ Z 2 1, Γ contains Z 2.
38 Proof in the complex-hyperbolic case Assume Γ is torsion-free. Since H 1 (Γ) 0 and Γ is a Kähler group, Theorem 1 Λ Γ Z 2 1. (Delzant s Alternative) Either Λ is f.g. or M = CH 2 /Γ holomorphically fibers over a Riemann surface. If M fibers then the image of the generic fiber group is f.g. but not f.p. (Kapovich). If Λ is f.g., then either it is not f.p. (and Γ is noncoherent) or Λ is f.p. and then is a PD(2) group (Hillman). Then Λ is a surface group (Eckmann Linnel Müller). But for any extension of the form 1 π 1 (S) Γ Z 2 1, Γ contains Z 2. (Essentially due to Birman Lubotzky McCarthy)
39 Proof in the complex-hyperbolic case Assume Γ is torsion-free. Since H 1 (Γ) 0 and Γ is a Kähler group, Theorem 1 Λ Γ Z 2 1. (Delzant s Alternative) Either Λ is f.g. or M = CH 2 /Γ holomorphically fibers over a Riemann surface. If M fibers then the image of the generic fiber group is f.g. but not f.p. (Kapovich). If Λ is f.g., then either it is not f.p. (and Γ is noncoherent) or Λ is f.p. and then is a PD(2) group (Hillman). Then Λ is a surface group (Eckmann Linnel Müller). But for any extension of the form 1 π 1 (S) Γ Z 2 1, Γ contains Z 2. (Essentially due to Birman Lubotzky McCarthy) Describe holomorphic fibration in the blown up complete quadrangle case if times permits.
40 Proof in the real-hyperbolic case Show that Γ contains a subgroup Λ isomorphic to A C B, where A, B are f.p. and H 1 (C) has infinite rank.
41 Proof in the real-hyperbolic case Show that Γ contains a subgroup Λ isomorphic to A C B, where A, B are f.p. and H 1 (C) has infinite rank. Then H 2 (Λ) has infinite rank and Λ is not even FP 2.
42 Proof in the real-hyperbolic case Show that Γ contains a subgroup Λ isomorphic to A C B, where A, B are f.p. and H 1 (C) has infinite rank. Then H 2 (Λ) has infinite rank and Λ is not even FP 2. More on this if time permits.
43 Remarks on SL(3, Z)
44 Remarks on SL(3, Z) Among finitely generated torsion-free infinite index non-solvable subgroups of SL(3, Z) we currently only know:
45 Remarks on SL(3, Z) Among finitely generated torsion-free infinite index non-solvable subgroups of SL(3, Z) we currently only know: 1. Free subgroups.
46 Remarks on SL(3, Z) Among finitely generated torsion-free infinite index non-solvable subgroups of SL(3, Z) we currently only know: 1. Free subgroups. 2. Groups Z 2 F k, where F k is free of rank k.
47 Remarks on SL(3, Z) Among finitely generated torsion-free infinite index non-solvable subgroups of SL(3, Z) we currently only know: 1. Free subgroups. 2. Groups Z 2 F k, where F k is free of rank k. 3. Surface subgroups.
48 Remarks on SL(3, Z) Among finitely generated torsion-free infinite index non-solvable subgroups of SL(3, Z) we currently only know: 1. Free subgroups. 2. Groups Z 2 F k, where F k is free of rank k. 3. Surface subgroups. As an example of the latter, consider Coxeter group T (3, 4, 4) or T (3, 6, 6). They are crystallographic (embed in GL(3, Z)) and contain surface subgroups of finite index.
49 Remarks on SL(3, Z) Among finitely generated torsion-free infinite index non-solvable subgroups of SL(3, Z) we currently only know: 1. Free subgroups. 2. Groups Z 2 F k, where F k is free of rank k. 3. Surface subgroups. As an example of the latter, consider Coxeter group T (3, 4, 4) or T (3, 6, 6). They are crystallographic (embed in GL(3, Z)) and contain surface subgroups of finite index. Note that they are all f.p.
50 Remarks on SL(3, Z) Among finitely generated torsion-free infinite index non-solvable subgroups of SL(3, Z) we currently only know: 1. Free subgroups. 2. Groups Z 2 F k, where F k is free of rank k. 3. Surface subgroups. As an example of the latter, consider Coxeter group T (3, 4, 4) or T (3, 6, 6). They are crystallographic (embed in GL(3, Z)) and contain surface subgroups of finite index. Note that they are all f.p. We do not even know if Z 2 Z embeds in SL(3, Z)!
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