On restricting subsets of bases in relatively free
groups
Lucas Sabalka
(joint with Dmytro Savchuk)
Binghamton University
Ithaca, NY
11 September 2011
Lucas Sabalka (Binghamton)
Subsets of Bases
11 September 2011
1 / 16
Motivation
Fn = ha1 , a2 , . . . , an i – free group of rank n
One potential analogue of Tits buildings/curve complex: splitting complex
To understand the geometry of the edge splitting graph, we introduced
i-length, a measure of complexity of words in Fn .
Lucas Sabalka (Binghamton)
Subsets of Bases
11 September 2011
2 / 16
Theorem (Whitehead)
If x is a power of a primitive element in Fn , then there is a cut vertex in Γ({x }).
bbccddb
a
(Draw Picture) a−1
Lucas Sabalka (Binghamton)
Subsets of Bases
b
c
d
b−1
c−1
d−1
11 September 2011
3 / 16
Theorem (Whitehead)
If x is a power of a primitive element in Fn , then there is a cut vertex in Γ({x }).
bbccddb
a
(Draw Picture) a−1
Lucas Sabalka (Binghamton)
Subsets of Bases
b
c
d
b−1
c−1
d−1
11 September 2011
3 / 16
No cut vertex, not primitive in F (b, c, d)
Lucas Sabalka (Binghamton)
Subsets of Bases
11 September 2011
4 / 16
Simple i-Length
Definition (simple i-length of w )
For w not involving ai , |w|simple
:= max t such that w = w1 . . . wt where the
i
Whitehead graph ΓA−{ai } (wi ) has no cut vertex.
conjugate-reduced i-length is similar, but takes conjugation into account
Lucas Sabalka (Binghamton)
Subsets of Bases
11 September 2011
4 / 16
Simple i-Length
Definition (simple i-length of w )
For w not involving ai , |w|simple
:= max t such that w = w1 . . . wt where the
i
Whitehead graph ΓA−{ai } (wi ) has no cut vertex.
conjugate-reduced i-length is similar, but takes conjugation into account
Lucas Sabalka (Binghamton)
Subsets of Bases
11 September 2011
4 / 16
One of our key Lemmas shows:
Lemma
For any basis x of Fn , any x ∈ x, and any subword w of αx x not involving ai ,
|w|cr
i = 0.
Corollary
If x is primitive in Fn and does not involve an , then its Whitehead graph with
respect to {a1 , . . . , an−1 } has a cut vertex.
bbccddb
(Draw Picture)
No cut vertex, so not primitive in F (a, b, c, d)
Question
Is every such x primitive in Fn−1 ?
Lucas Sabalka (Binghamton)
Subsets of Bases
11 September 2011
5 / 16
One of our key Lemmas shows:
Lemma
For any basis x of Fn , any x ∈ x, and any subword w of αx x not involving ai ,
|w|cr
i = 0.
Corollary
If x is primitive in Fn and does not involve an , then its Whitehead graph with
respect to {a1 , . . . , an−1 } has a cut vertex.
bbccddb
(Draw Picture)
No cut vertex, so not primitive in F (a, b, c, d)
Question
Is every such x primitive in Fn−1 ?
Lucas Sabalka (Binghamton)
Subsets of Bases
11 September 2011
5 / 16
One of our key Lemmas shows:
Lemma
For any basis x of Fn , any x ∈ x, and any subword w of αx x not involving ai ,
|w|cr
i = 0.
Corollary
If x is primitive in Fn and does not involve an , then its Whitehead graph with
respect to {a1 , . . . , an−1 } has a cut vertex.
bbccddb
(Draw Picture)
No cut vertex, so not primitive in F (a, b, c, d)
Question
Is every such x primitive in Fn−1 ?
Lucas Sabalka (Binghamton)
Subsets of Bases
11 September 2011
5 / 16
One of our key Lemmas shows:
Lemma
For any basis x of Fn , any x ∈ x, and any subword w of αx x not involving ai ,
|w|cr
i = 0.
Corollary
If x is primitive in Fn and does not involve an , then its Whitehead graph with
respect to {a1 , . . . , an−1 } has a cut vertex.
bbccddb
(Draw Picture)
No cut vertex, so not primitive in F (a, b, c, d)
Question
Is every such x primitive in Fn−1 ?
Lucas Sabalka (Binghamton)
Subsets of Bases
11 September 2011
5 / 16
Or more generally:
Question
Is every primitive subset of Fn that does not involve an primitive in Fn−1 ?
Question (Sapir)
What about relatively free groups?
Lucas Sabalka (Binghamton)
Subsets of Bases
11 September 2011
6 / 16
Or more generally:
Question
Is every primitive subset of Fn that does not involve an primitive in Fn−1 ?
Question (Sapir)
What about relatively free groups?
Lucas Sabalka (Binghamton)
Subsets of Bases
11 September 2011
6 / 16
More General Notation
F = F (A) – free group of rank n ≥ 2 with basis A = {a1 , . . . , an }
V – a fully invariant subgroup of F
Cases of interest:
V = 1 is the trivial subgroup,
and F /V = F is a free group
V = F (k ) is a term of the derived series of F ,
and F /V is a free solvable group of derived length k
V = γc (F ) is a term of the lower central series of F ,
and F /V is a free nilpotent group of class c
Lucas Sabalka (Binghamton)
Subsets of Bases
11 September 2011
7 / 16
More General Notation
F = F (A) – free group of rank n ≥ 2 with basis A = {a1 , . . . , an }
V – a fully invariant subgroup of F
Cases of interest:
V = 1 is the trivial subgroup,
and F /V = F is a free group
V = F (k ) is a term of the derived series of F ,
and F /V is a free solvable group of derived length k
V = γc (F ) is a term of the lower central series of F ,
and F /V is a free nilpotent group of class c
Lucas Sabalka (Binghamton)
Subsets of Bases
11 September 2011
7 / 16
Definition
A subset S ⊂ F is primitive in F mod V if the corresponding set of cosets SV
of V can be extended to a basis of F /V .
Let S ⊂ F be primitive mod V .
Assume an is not used to express any s ∈ S:
S ⊂ F̂ := F (A − {an })
Consider V̂ = V ∩ F̂ , a fully invariant subgroup of F̂ .
Question
When is S also primitive in F̂ mod V̂ ?
Lucas Sabalka (Binghamton)
Subsets of Bases
11 September 2011
8 / 16
Definition
A subset S ⊂ F is primitive in F mod V if the corresponding set of cosets SV
of V can be extended to a basis of F /V .
Let S ⊂ F be primitive mod V .
Assume an is not used to express any s ∈ S:
S ⊂ F̂ := F (A − {an })
Consider V̂ = V ∩ F̂ , a fully invariant subgroup of F̂ .
Question
When is S also primitive in F̂ mod V̂ ?
Lucas Sabalka (Binghamton)
Subsets of Bases
11 September 2011
8 / 16
Definition
A subset S ⊂ F is primitive in F mod V if the corresponding set of cosets SV
of V can be extended to a basis of F /V .
Let S ⊂ F be primitive mod V .
Assume an is not used to express any s ∈ S:
S ⊂ F̂ := F (A − {an })
Consider V̂ = V ∩ F̂ , a fully invariant subgroup of F̂ .
Question
When is S also primitive in F̂ mod V̂ ?
Lucas Sabalka (Binghamton)
Subsets of Bases
11 September 2011
8 / 16
Definition
A subset S ⊂ F is primitive in F mod V if the corresponding set of cosets SV
of V can be extended to a basis of F /V .
Let S ⊂ F be primitive mod V .
Assume an is not used to express any s ∈ S:
S ⊂ F̂ := F (A − {an })
Consider V̂ = V ∩ F̂ , a fully invariant subgroup of F̂ .
Question
When is S also primitive in F̂ mod V̂ ?
Lucas Sabalka (Binghamton)
Subsets of Bases
11 September 2011
8 / 16
Our Results
Theorem (SS)
Let S ⊂ F̂ be primitive in F mod V . Then S is also primitive in F̂ mod V̂ if:
1
2
F /V = F is free
F /V is free abelian
3
F /V is free metabelian
4
F /V is free nilpotent of class c and |S| ≤ n − c
F /V is free nilpotent of class 2
5
(thanks to referee)
Claim
1
Result still holds if F̂ misses more than 1 element.
2
Result holds when F /V is free solvable and |S| = n − 1.
Lucas Sabalka (Binghamton)
Subsets of Bases
11 September 2011
9 / 16
Our Results
Theorem (SS)
Let S ⊂ F̂ be primitive in F mod V . Then S is also primitive in F̂ mod V̂ if:
1
2
F /V = F is free
F /V is free abelian
3
F /V is free metabelian
4
F /V is free nilpotent of class c and |S| ≤ n − c
F /V is free nilpotent of class 2
5
(thanks to referee)
Claim
1
Result still holds if F̂ misses more than 1 element.
2
Result holds when F /V is free solvable and |S| = n − 1.
Lucas Sabalka (Binghamton)
Subsets of Bases
11 September 2011
9 / 16
Proof for free groups: Fox Calculus
For each j = 1, 2, . . . , n define the free Fox derivative Dj : ZFn → ZFn
recursively by
Dj (aj ) = 1,
Dj (ai ) = 0, i 6= j
and
Dj (uv ) = Dj (u) + uDj (v ) for all u, v ∈ Fn .
Example
For a2 ba ∈ F (a, b) we have
Da (a2 b · a)
Lucas Sabalka (Binghamton)
=
=
=
=
Da (a2 · b) + a2 b · Da (a)
Da (a · a) + a2 · Da (b) + a2 b
Da (a) + a · Da (a) + a2 b
1 + a + a2 b
Subsets of Bases
11 September 2011
10 / 16
Proof for free groups: Fox Calculus
For each j = 1, 2, . . . , n define the free Fox derivative Dj : ZFn → ZFn
recursively by
Dj (aj ) = 1,
Dj (ai ) = 0, i 6= j
and
Dj (uv ) = Dj (u) + uDj (v ) for all u, v ∈ Fn .
Example
For a2 ba ∈ F (a, b) we have
Da (a2 b · a)
Lucas Sabalka (Binghamton)
=
=
=
=
Da (a2 · b) + a2 b · Da (a)
Da (a · a) + a2 · Da (b) + a2 b
Da (a) + a · Da (a) + a2 b
1 + a + a2 b
Subsets of Bases
11 September 2011
10 / 16
Proof for free groups: Fox Calculus
For each j = 1, 2, . . . , n define the free Fox derivative Dj : ZFn → ZFn
recursively by
Dj (aj ) = 1,
Dj (ai ) = 0, i 6= j
and
Dj (uv ) = Dj (u) + uDj (v ) for all u, v ∈ Fn .
Example
For a2 ba ∈ F (a, b) we have
Da (a2 b · a)
Lucas Sabalka (Binghamton)
=
=
=
=
Da (a2 · b) + a2 b · Da (a)
Da (a · a) + a2 · Da (b) + a2 b
Da (a) + a · Da (a) + a2 b
1 + a + a2 b
Subsets of Bases
11 September 2011
10 / 16
Theorem (Birman (73) ⇒, Umirbaev (94) ⇐)
A subset {x1 , . . . , xk } is primitive in Fn if and only if the k × n Jacobian matrix


D1 (x1 ) · · · Dn (x1 )


..
..
..
J=

.
.
.
D1 (xk ) · · · Dn (xk )
is right invertible in ZFn .
Lucas Sabalka (Binghamton)
Subsets of Bases
11 September 2011
11 / 16
Theorem
Let S = {x1 , x2 , . . . , xk } ⊂ F̂ be primitive in F . Then S is primitive in F̂ .
Proof:
S is primitive in F ⇒ Jacobian J of S is right invertible.
So ∃ P = (pjl ) with pjl ∈ ZFn satisfying
JP = Ik
No an in S ⇒
no an in J
n-th column of J consists of zeros.
Write each entry of P as
pjl = qjl + rjl
where
each term in qjl involves an
each term in rjl does not involves an
Lucas Sabalka (Binghamton)
Subsets of Bases
11 September 2011
12 / 16
Theorem
Let S = {x1 , x2 , . . . , xk } ⊂ F̂ be primitive in F . Then S is primitive in F̂ .
Proof:
S is primitive in F ⇒ Jacobian J of S is right invertible.
So ∃ P = (pjl ) with pjl ∈ ZFn satisfying
JP = Ik
No an in S ⇒
no an in J
n-th column of J consists of zeros.
Write each entry of P as
pjl = qjl + rjl
where
each term in qjl involves an
each term in rjl does not involves an
Lucas Sabalka (Binghamton)
Subsets of Bases
11 September 2011
12 / 16
Theorem
Let S = {x1 , x2 , . . . , xk } ⊂ F̂ be primitive in F . Then S is primitive in F̂ .
Proof:
S is primitive in F ⇒ Jacobian J of S is right invertible.
So ∃ P = (pjl ) with pjl ∈ ZFn satisfying
JP = Ik
No an in S ⇒
no an in J
n-th column of J consists of zeros.
Write each entry of P as
pjl = qjl + rjl
where
each term in qjl involves an
each term in rjl does not involves an
Lucas Sabalka (Binghamton)
Subsets of Bases
11 September 2011
12 / 16
Theorem
Let S = {x1 , x2 , . . . , xk } ⊂ F̂ be primitive in F . Then S is primitive in F̂ .
Proof:
S is primitive in F ⇒ Jacobian J of S is right invertible.
So ∃ P = (pjl ) with pjl ∈ ZFn satisfying
JP = Ik
No an in S ⇒
no an in J
n-th column of J consists of zeros.
Write each entry of P as
pjl = qjl + rjl
where
each term in qjl involves an
each term in rjl does not involves an
Lucas Sabalka (Binghamton)
Subsets of Bases
11 September 2011
12 / 16
Then for matrices Q = (qjl ) and R = (rjl )
P =Q+R
JQ + JR = JP = Ik
But each entry of JQ is either 0 or involves an , so JQ = 0, and
JR = Ik
Let J̃ and R̃ be the matrices obtained from J and R by deleting n-th row and
n-th column correspondingly. Then
J̃ is the Jacobian matrix of the set S seen as a subset of Fn−1
R̃ is a matrix over ZFn−1 .
Also
J̃ R̃ = Ik .
By Umirbaev’s criterion the set S is a subset of the basis of F̂ .
Lucas Sabalka (Binghamton)
Subsets of Bases
11 September 2011
13 / 16
Then for matrices Q = (qjl ) and R = (rjl )
P =Q+R
JQ + JR = JP = Ik
But each entry of JQ is either 0 or involves an , so JQ = 0, and
JR = Ik
Let J̃ and R̃ be the matrices obtained from J and R by deleting n-th row and
n-th column correspondingly. Then
J̃ is the Jacobian matrix of the set S seen as a subset of Fn−1
R̃ is a matrix over ZFn−1 .
Also
J̃ R̃ = Ik .
By Umirbaev’s criterion the set S is a subset of the basis of F̂ .
Lucas Sabalka (Binghamton)
Subsets of Bases
11 September 2011
13 / 16
Then for matrices Q = (qjl ) and R = (rjl )
P =Q+R
JQ + JR = JP = Ik
But each entry of JQ is either 0 or involves an , so JQ = 0, and
JR = Ik
Let J̃ and R̃ be the matrices obtained from J and R by deleting n-th row and
n-th column correspondingly. Then
J̃ is the Jacobian matrix of the set S seen as a subset of Fn−1
R̃ is a matrix over ZFn−1 .
Also
J̃ R̃ = Ik .
By Umirbaev’s criterion the set S is a subset of the basis of F̂ .
Lucas Sabalka (Binghamton)
Subsets of Bases
11 September 2011
13 / 16
Then for matrices Q = (qjl ) and R = (rjl )
P =Q+R
JQ + JR = JP = Ik
But each entry of JQ is either 0 or involves an , so JQ = 0, and
JR = Ik
Let J̃ and R̃ be the matrices obtained from J and R by deleting n-th row and
n-th column correspondingly. Then
J̃ is the Jacobian matrix of the set S seen as a subset of Fn−1
R̃ is a matrix over ZFn−1 .
Also
J̃ R̃ = Ik .
By Umirbaev’s criterion the set S is a subset of the basis of F̂ .
Lucas Sabalka (Binghamton)
Subsets of Bases
11 September 2011
13 / 16
Then for matrices Q = (qjl ) and R = (rjl )
P =Q+R
JQ + JR = JP = Ik
But each entry of JQ is either 0 or involves an , so JQ = 0, and
JR = Ik
Let J̃ and R̃ be the matrices obtained from J and R by deleting n-th row and
n-th column correspondingly. Then
J̃ is the Jacobian matrix of the set S seen as a subset of Fn−1
R̃ is a matrix over ZFn−1 .
Also
J̃ R̃ = Ik .
By Umirbaev’s criterion the set S is a subset of the basis of F̂ .
Lucas Sabalka (Binghamton)
Subsets of Bases
11 September 2011
13 / 16
Then for matrices Q = (qjl ) and R = (rjl )
P =Q+R
JQ + JR = JP = Ik
But each entry of JQ is either 0 or involves an , so JQ = 0, and
JR = Ik
Let J̃ and R̃ be the matrices obtained from J and R by deleting n-th row and
n-th column correspondingly. Then
J̃ is the Jacobian matrix of the set S seen as a subset of Fn−1
R̃ is a matrix over ZFn−1 .
Also
J̃ R̃ = Ik .
By Umirbaev’s criterion the set S is a subset of the basis of F̂ .
Lucas Sabalka (Binghamton)
Subsets of Bases
11 September 2011
13 / 16
Then for matrices Q = (qjl ) and R = (rjl )
P =Q+R
JQ + JR = JP = Ik
But each entry of JQ is either 0 or involves an , so JQ = 0, and
JR = Ik
Let J̃ and R̃ be the matrices obtained from J and R by deleting n-th row and
n-th column correspondingly. Then
J̃ is the Jacobian matrix of the set S seen as a subset of Fn−1
R̃ is a matrix over ZFn−1 .
Also
J̃ R̃ = Ik .
By Umirbaev’s criterion the set S is a subset of the basis of F̂ .
Lucas Sabalka (Binghamton)
Subsets of Bases
11 September 2011
13 / 16
Then for matrices Q = (qjl ) and R = (rjl )
P =Q+R
JQ + JR = JP = Ik
But each entry of JQ is either 0 or involves an , so JQ = 0, and
JR = Ik
Let J̃ and R̃ be the matrices obtained from J and R by deleting n-th row and
n-th column correspondingly. Then
J̃ is the Jacobian matrix of the set S seen as a subset of Fn−1
R̃ is a matrix over ZFn−1 .
Also
J̃ R̃ = Ik .
By Umirbaev’s criterion the set S is a subset of the basis of F̂ .
Lucas Sabalka (Binghamton)
Subsets of Bases
11 September 2011
13 / 16
The Metabelian Case
The Same
Mn = F /F ′′ – free metabelian group
An = F /F ′ – free abelian group
Free Fox derivative Dj induces
dj : ZMn → ZAn
We have
dj (aj ) = 1,
dj (ai ) = 0, i 6= j
and
dj (uv ) = dj (u) + π(u)dj (v ) for all u, v ∈ Mn ,
where π : Mn → An is the canonical abelianization epimorphism.
Lucas Sabalka (Binghamton)
Subsets of Bases
11 September 2011
14 / 16
The Metabelian Case
The Same
Mn = F /F ′′ – free metabelian group
An = F /F ′ – free abelian group
Free Fox derivative Dj induces
dj : ZMn → ZAn
We have
dj (aj ) = 1,
dj (ai ) = 0, i 6= j
and
dj (uv ) = dj (u) + π(u)dj (v ) for all u, v ∈ Mn ,
where π : Mn → An is the canonical abelianization epimorphism.
Lucas Sabalka (Binghamton)
Subsets of Bases
11 September 2011
14 / 16
The Metabelian Case
The Same
Mn = F /F ′′ – free metabelian group
An = F /F ′ – free abelian group
Free Fox derivative Dj induces
dj : ZMn → ZAn
We have
dj (aj ) = 1,
dj (ai ) = 0, i 6= j
and
dj (uv ) = dj (u) + π(u)dj (v ) for all u, v ∈ Mn ,
where π : Mn → An is the canonical abelianization epimorphism.
Lucas Sabalka (Binghamton)
Subsets of Bases
11 September 2011
14 / 16
The Metabelian Case
The Same
Mn = F /F ′′ – free metabelian group
An = F /F ′ – free abelian group
Free Fox derivative Dj induces
dj : ZMn → ZAn
We have
dj (aj ) = 1,
dj (ai ) = 0, i 6= j
and
dj (uv ) = dj (u) + π(u)dj (v ) for all u, v ∈ Mn ,
where π : Mn → An is the canonical abelianization epimorphism.
Lucas Sabalka (Binghamton)
Subsets of Bases
11 September 2011
14 / 16
The Metabelian Case
The Same
Mn = F /F ′′ – free metabelian group
An = F /F ′ – free abelian group
Free Fox derivative Dj induces
dj : ZMn → ZAn
We have
dj (aj ) = 1,
dj (ai ) = 0, i 6= j
and
dj (uv ) = dj (u) + π(u)dj (v ) for all u, v ∈ Mn ,
where π : Mn → An is the canonical abelianization epimorphism.
Lucas Sabalka (Binghamton)
Subsets of Bases
11 September 2011
14 / 16
The Metabelian Case
The Same
Mn = F /F ′′ – free metabelian group
An = F /F ′ – free abelian group
Free Fox derivative Dj induces
dj : ZMn → ZAn
We have
dj (aj ) = 1,
dj (ai ) = 0, i 6= j
and
dj (uv ) = dj (u) + π(u)dj (v ) for all u, v ∈ Mn ,
where π : Mn → An is the canonical abelianization epimorphism.
Lucas Sabalka (Binghamton)
Subsets of Bases
11 September 2011
14 / 16
Theorem (Timoshenko (89,92), Roman’kov (91))
A subset {x1 , . . . , xk } is primitive in Mn if and only if
J := dj (xl ) , 1 ≤ l ≤ k , 1 ≤ j ≤ n is right invertible in the integral ring ZAn .
Corollary
Let S ⊂ F̂ be primitive in F mod F ′′ . Then S is primitive in F̂ mod F̂ ′′ .
Question
Can we extend this technique to free solvable groups? (must deal with wild
automorphisms)
Lucas Sabalka (Binghamton)
Subsets of Bases
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15 / 16
Theorem (Timoshenko (89,92), Roman’kov (91))
A subset {x1 , . . . , xk } is primitive in Mn if and only if
J := dj (xl ) , 1 ≤ l ≤ k , 1 ≤ j ≤ n is right invertible in the integral ring ZAn .
Corollary
Let S ⊂ F̂ be primitive in F mod F ′′ . Then S is primitive in F̂ mod F̂ ′′ .
Question
Can we extend this technique to free solvable groups? (must deal with wild
automorphisms)
Lucas Sabalka (Binghamton)
Subsets of Bases
11 September 2011
15 / 16
Theorem (Timoshenko (89,92), Roman’kov (91))
A subset {x1 , . . . , xk } is primitive in Mn if and only if
J := dj (xl ) , 1 ≤ l ≤ k , 1 ≤ j ≤ n is right invertible in the integral ring ZAn .
Corollary
Let S ⊂ F̂ be primitive in F mod F ′′ . Then S is primitive in F̂ mod F̂ ′′ .
Question
Can we extend this technique to free solvable groups? (must deal with wild
automorphisms)
Lucas Sabalka (Binghamton)
Subsets of Bases
11 September 2011
15 / 16
The Nilpotent Case
γc (F ) – c-th term in the lower central series of F .
γ1 (F ) = F
γn+1 (F ) = [γn (F ), F ]
F /γc+1 (F ) – free nilpotent group of class c.
Theorem (Gupta-Gupta, 92)
Let F /γc+1 (F ) be the free nilpotent group of class c ≥ 2. If S is primitive in F
mod γc+1 (F ) and k ≤ n − c + 1, then S lifts to a set S̃ which is primitive in F .
Theorem (SS)
Moreover, if k ≤ n − c, and S ⊂ F̂ , then S̃ can be chosen to be inside F̂ .
Corollary
Let S ⊂ F̂ be primitive in F mod γc+1 (F ). If n > 2 and k ≤ n − c then S is
primitive in F̂ mod γc+1 (F̂ ).
Lucas Sabalka (Binghamton)
Subsets of Bases
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16 / 16
The Nilpotent Case
γc (F ) – c-th term in the lower central series of F .
γ1 (F ) = F
γn+1 (F ) = [γn (F ), F ]
F /γc+1 (F ) – free nilpotent group of class c.
Theorem (Gupta-Gupta, 92)
Let F /γc+1 (F ) be the free nilpotent group of class c ≥ 2. If S is primitive in F
mod γc+1 (F ) and k ≤ n − c + 1, then S lifts to a set S̃ which is primitive in F .
Theorem (SS)
Moreover, if k ≤ n − c, and S ⊂ F̂ , then S̃ can be chosen to be inside F̂ .
Corollary
Let S ⊂ F̂ be primitive in F mod γc+1 (F ). If n > 2 and k ≤ n − c then S is
primitive in F̂ mod γc+1 (F̂ ).
Lucas Sabalka (Binghamton)
Subsets of Bases
11 September 2011
16 / 16
The Nilpotent Case
γc (F ) – c-th term in the lower central series of F .
γ1 (F ) = F
γn+1 (F ) = [γn (F ), F ]
F /γc+1 (F ) – free nilpotent group of class c.
Theorem (Gupta-Gupta, 92)
Let F /γc+1 (F ) be the free nilpotent group of class c ≥ 2. If S is primitive in F
mod γc+1 (F ) and k ≤ n − c + 1, then S lifts to a set S̃ which is primitive in F .
Theorem (SS)
Moreover, if k ≤ n − c, and S ⊂ F̂ , then S̃ can be chosen to be inside F̂ .
Corollary
Let S ⊂ F̂ be primitive in F mod γc+1 (F ). If n > 2 and k ≤ n − c then S is
primitive in F̂ mod γc+1 (F̂ ).
Lucas Sabalka (Binghamton)
Subsets of Bases
11 September 2011
16 / 16
The Nilpotent Case
γc (F ) – c-th term in the lower central series of F .
γ1 (F ) = F
γn+1 (F ) = [γn (F ), F ]
F /γc+1 (F ) – free nilpotent group of class c.
Theorem (Gupta-Gupta, 92)
Let F /γc+1 (F ) be the free nilpotent group of class c ≥ 2. If S is primitive in F
mod γc+1 (F ) and k ≤ n − c + 1, then S lifts to a set S̃ which is primitive in F .
Theorem (SS)
Moreover, if k ≤ n − c, and S ⊂ F̂ , then S̃ can be chosen to be inside F̂ .
Corollary
Let S ⊂ F̂ be primitive in F mod γc+1 (F ). If n > 2 and k ≤ n − c then S is
primitive in F̂ mod γc+1 (F̂ ).
Lucas Sabalka (Binghamton)
Subsets of Bases
11 September 2011
16 / 16
The Nilpotent Case
γc (F ) – c-th term in the lower central series of F .
γ1 (F ) = F
γn+1 (F ) = [γn (F ), F ]
F /γc+1 (F ) – free nilpotent group of class c.
Theorem (Gupta-Gupta, 92)
Let F /γc+1 (F ) be the free nilpotent group of class c ≥ 2. If S is primitive in F
mod γc+1 (F ) and k ≤ n − c + 1, then S lifts to a set S̃ which is primitive in F .
Theorem (SS)
Moreover, if k ≤ n − c, and S ⊂ F̂ , then S̃ can be chosen to be inside F̂ .
Corollary
Let S ⊂ F̂ be primitive in F mod γc+1 (F ). If n > 2 and k ≤ n − c then S is
primitive in F̂ mod γc+1 (F̂ ).
Lucas Sabalka (Binghamton)
Subsets of Bases
11 September 2011
16 / 16
The Nilpotent Case
γc (F ) – c-th term in the lower central series of F .
γ1 (F ) = F
γn+1 (F ) = [γn (F ), F ]
F /γc+1 (F ) – free nilpotent group of class c.
Theorem (Gupta-Gupta, 92)
Let F /γc+1 (F ) be the free nilpotent group of class c ≥ 2. If S is primitive in F
mod γc+1 (F ) and k ≤ n − c + 1, then S lifts to a set S̃ which is primitive in F .
Theorem (SS)
Moreover, if k ≤ n − c, and S ⊂ F̂ , then S̃ can be chosen to be inside F̂ .
Corollary
Let S ⊂ F̂ be primitive in F mod γc+1 (F ). If n > 2 and k ≤ n − c then S is
primitive in F̂ mod γc+1 (F̂ ).
Lucas Sabalka (Binghamton)
Subsets of Bases
11 September 2011
16 / 16