Krull dimension for limit groups
Notes taken by W. Malone
Larsen Louder
Abstract
When fields are replaced by free groups, the notion of Krull dimension from
the theory of commutative rings has a natural interpretation as the supremum
of lengths of chains of proper epimorphisms of limit groups beginning with a
free group of fixed rank. I will sketch parts of the proof that it is finite and
give examples of the techniques and some inherent difficulties.
In this talk F will always be a free group.
Commutative algebra
k[x1 , . . . , xn ], k
f (xi )
I =< fi (xi ) >
V (I) = {x|f (xi ) = 0 ∀f ∈ I}
k[V ] = k[x1 , . . . , xn ]/I
1−1
V ←→
√ {k[V ] → k}
I = I radical ideal
I prime
Groups
< x1 , . . . , xn >, F
ω(xi )
Σ = ωj (xi ) VΣ = {x ∈ F n |ωj (xi ) = 1}
GΣ =< x1 , . . . , xn > /Σ
1−1
VΣ ←→ {GΣ → F }
GΣ is residually free
GΣ is ω-residually free
Krull Dimension
Definition 1. G is ω-residually free if for all S ⊂ G with S finite there exists f :
G → F such that f |S is injective.
If not ∃a1 , . . . an such that ∀f : G → F f (ai ) = 1 for some i. We should think G
is ω-residually free if and only if Hom(G, F ) is irreducible.
Theorem 2. (Sela) G not ω-residually free then there exists a finite collection of
ηi : G → Li with Li ω-residually free such that every homomorphism f : G → F
factors through some Li .
Definition 3. The Krull dimension is the sup of lengths of chains of prime ideals.
k n ⊃ V1 ⊃ V2 ⊂ · · · ⊃ Vk ! h[x1 , . . . ] ⊃ Pn ⊃ Pn−1 ⊃ · · · ⊃ P1
where the Vi are irreducible.
1
2
L. 1
..
Ls
η1
fe
ηs
G
f
F
Theorem 4. (Louder) There exists a function D(n) such that if Fn =< x1 , . . . , xn >
L1 L2 · · · Lk−1 Lk 1 is a sequence of proper epimorphisms of limit
groups then k ≤ D(n) .
Definition 5. G is a finitely generated group and fn : G → F is a sequence of
homomorphisms is convergent if ∀g ∈ G there exists n0 (g) such that ∀n > n0 fn (g) 6=
1 or ∀n > n0 fn (g) = 1. The stable kernal of fn is Ker
−−→(fn ) = {g|fn (g) = 1, n >
n0 (g)}. G/Ker
−−→(fn ) = L is a limit group.
Facts:
• L is a limit group if and only if L is ω-residually free.
• finitely presented, coherent, subgroups are limit groups
• abelian subgroups are finitely generated and free.
• maximum abelian subgroups are malnormal
• satisfy commutative transitivity
• Limit groups have abelian JSJ decomposition (Sela).
Definition 6. Let L be a limit group that is freely indecomposable which is not a
closed surface group or abelian. Then L has an abelian JSJ over rigid, quadradically
hanging, or abelian vertex groups, and the edge groups are finitely generated abelian
groups.
11 0
00
1
In the JSJ decomposition every one edge splitting of L over some edge group E is
visible: collapse all but one edge or cut along a simple closed curve on a QH vertex
group. A special kind of example of a homomorphism of a limit group is “strict”.
3
M
.. 1
.
Ms
ϕ ∈ M od(L)
ϕ
L
fe
F
f
Theorem 7. (Sela)
Ker
The Mi are proper limit group quotients. Consider fn : L → F such that →
(fn ) = 1 “converges to L”. Without loss of generality fn = gn ◦η◦ϕn where η : L → M1
and gn : Mn → F .
• η embeds rigid vertex groups and edge groups
• η(quadradically hanging vertex group) is non-abelian.
• Other stuff
π
Definition 8. (“strict”) L1 → L2
1. Strict in the above sense when restricted to freely indecomposable free factors.
2. Non-abelian image
3. Does not kill non-cyclic abelian freely indecomposable free factors.
Ker
Alternatively we could require that for all fn : L2 → F with → (f1 ) = 1 then
there exists ϕn ∈ M od(L1 ) such that fn ◦ π ◦ ϕn converges to L1 .
Compatibility verses strictness
Theorem 9. If
π
π
fn → L1 →1 L2 →2 · · · → Lk
with πi strict and proper then k ≤ 3n.
Definition 10. L1 L2 is “almost an isomorphism” if for all strict S : M → L2
making the diagram commute then S is an isomorphism.
M
S
L1
L2
4
Theorem 11. There exists a complexity “SC” such that if L1 → L2 is an almost
isomorphism then SC(L1 ) ≤ SC(L2 ). If “=” holds then L1 and L2 have the “same”
Grusko decomposition and the freely indecomposable free factors of L1 , L2 have the
“same” JSJ. Moreover SC takes boundedly many values on limit groups of fixed rank
(minimal number of generators).
Homeo
‘‘Same JSJ’’
*
*
Example 12. Let the figure below be such that Ker
−−→→(fn ) = 1, K = Ker
−−→→(fn ◦ π ◦
τ
= L1
π
L2
fn
F
∆
τ m(n) ), and M = L1 /K. We desire a map M → L2 such that
L1
τ
L2
M
τe
M
If this is the case then we would like to show that the map L1 → M respects the
original decomposition ∆.
5
= L1
π
L2
F
fn
where fn ◦ π ◦ τ M (n) , π() = nonabelian and π(= (edge)) = nontrivial. Choose
M (n) huge then
fn ◦ π ◦ τ M (n)
L1 y Cayley graph of F.
Take a limit of these actions to get an action L1 /Ker
−−→(fn ) = M y T on a simplicial
R-tree T . This gives a 1-edge splitting of M and the images of the vertex groups of
L1 act elliptically.
τ
L1
This is roughly how to create compatibility.
τe
M