Quantifying residual finiteness in groups and algebras

Be’eri Greenfeld

Department of MathematicsBar Ilan University, Israel

Groups, Rings and Associated Structures 2019

Be’eri Greenfeld (BIU) Residual girth of groups and algebras Spa 2019 1 / 11

Residual girth of groups

G = 〈S〉 - finitely generated residually finite group

Can we measure its residual finiteness?

=What is the smallest homomorphic image in which the n-ball of theCayley graph embeds?

Definition (Residual girth)

resG (n) = min{[G : N]|N / G s.t. N ∩ Sn = {e}}

This is defined independently of S up to a natural equivalence of functionsf ∼ g (=f (n) ≤ Cg(Dn) ≤ C ′f (D ′n))

Be’eri Greenfeld (BIU) Residual girth of groups and algebras Spa 2019 2 / 11

Residual girth of groups

G = 〈S〉 - finitely generated residually finite group

Can we measure its residual finiteness?

=What is the smallest homomorphic image in which the n-ball of theCayley graph embeds?

Definition (Residual girth)

resG (n) = min{[G : N]|N / G s.t. N ∩ Sn = {e}}

This is defined independently of S up to a natural equivalence of functionsf ∼ g (=f (n) ≤ Cg(Dn) ≤ C ′f (D ′n))

Be’eri Greenfeld (BIU) Residual girth of groups and algebras Spa 2019 2 / 11

Residual girth of groups

G = 〈S〉 - finitely generated residually finite group

Can we measure its residual finiteness?

=What is the smallest homomorphic image in which the n-ball of theCayley graph embeds?

Definition (Residual girth)

resG (n) = min{[G : N]|N / G s.t. N ∩ Sn = {e}}

This is defined independently of S up to a natural equivalence of functionsf ∼ g (=f (n) ≤ Cg(Dn) ≤ C ′f (D ′n))

Be’eri Greenfeld (BIU) Residual girth of groups and algebras Spa 2019 2 / 11

Residual girth of groups

Some results:

resG (n) ≈ poly(n) iff G virt. nilp.

resG (n) � exp(n) for linear groups

resG (n) is computable if G is finitely presented or has slovable wordnormal fin. ind. subgroup membership problems

What does residual girth growth reflect? (cf. word growth?)

In general, resG (n) can grow arbitrarily fast

For branch groups =⇒ resG (n) reflects contracting properties(Grigorchuk’s group: exponential; Gupta-Sidki: super-exponential...)

Among nilpotent groups: residual girth ≈ word growth only forvirtually abelian groups (lower central series; terraced filtrations)

Be’eri Greenfeld (BIU) Residual girth of groups and algebras Spa 2019 3 / 11

Residual girth of groups

Some results:

resG (n) ≈ poly(n) iff G virt. nilp.

resG (n) � exp(n) for linear groups

resG (n) is computable if G is finitely presented or has slovable wordnormal fin. ind. subgroup membership problems

What does residual girth growth reflect? (cf. word growth?)

In general, resG (n) can grow arbitrarily fast

For branch groups =⇒ resG (n) reflects contracting properties(Grigorchuk’s group: exponential; Gupta-Sidki: super-exponential...)

Among nilpotent groups: residual girth ≈ word growth only forvirtually abelian groups (lower central series; terraced filtrations)

Be’eri Greenfeld (BIU) Residual girth of groups and algebras Spa 2019 3 / 11

Residual girth of algebras

Fix a base field F . Suppose R = F 〈V 〉 is a finitely generated, residuallyfinite dimensional associative algebra (V is a finite dimensional generatingsubspace containing 1).Idea: residual girth of R measures how efficiently V n can be detected infinite dimensional quotients of R.

Definition

resG (n) = min{[G : N]|N / G s.t. N ∩ Sn = {e}}

Be’eri Greenfeld (BIU) Residual girth of groups and algebras Spa 2019 4 / 11

Residual girth of algebras

Fix a base field F . Suppose R = F 〈V 〉 is a finitely generated, residuallyfinite dimensional associative algebra (V is a finite dimensional generatingsubspace containing 1).Idea: residual girth of R measures how efficiently V n can be detected infinite dimensional quotients of R.

Definition

resR(n) = min{dimF (R/I )|I / R s.t. I ∩ V n = 0}

(As before, well defined indpt. of V up to natural growth functionsequivalence.)The growth again bounds from below: γR(n) = dimF V n � resR(n).

Be’eri Greenfeld (BIU) Residual girth of groups and algebras Spa 2019 5 / 11

Residual girth of algebras

Fix a base field F . Suppose R = F 〈V 〉 is a finitely generated, residuallyfinite dimensional associative algebra (V is a finite dimensional generatingsubspace containing 1).Idea: residual girth of R measures how efficiently V n can be detected infinite dimensional quotients of R.

Definition

resR(n) = min{dimF (R/I )|I / R s.t. I ∩ V n = 0}

(As before, well defined indpt. of V up to natural growth functionsequivalence.)The growth again bounds from below: γR(n) = dimF V n � resR(n).

Be’eri Greenfeld (BIU) Residual girth of groups and algebras Spa 2019 5 / 11

Comparing groups and algebras

Consider group algebras: resG (n) vs. resF [G ](n)?

Evidently: resF [G ](n) � resG (n) (normal subgroups give rise to ideals).

Question

Is resG (n) ∼ resF [G ](n)?

Motivation: Can n-Balls be more efficiently detected if we usering-theoretic homomorphic images? ‘Quantitative ideal congruenceproperty’ - do ideals of F [G ] contain ‘large enough’ relative augmentationideals?Answer: No. H = 〈x , y |[x , y ] central〉 Heisenberg group.It was computed that resH(n) ∼ n6 (recall that γH(n) ∼ n4).However...

Be’eri Greenfeld (BIU) Residual girth of groups and algebras Spa 2019 6 / 11

Comparing groups and algebras

Consider group algebras: resG (n) vs. resF [G ](n)?Evidently: resF [G ](n) � resG (n) (normal subgroups give rise to ideals).

Question

Is resG (n) ∼ resF [G ](n)?

Motivation: Can n-Balls be more efficiently detected if we usering-theoretic homomorphic images? ‘Quantitative ideal congruenceproperty’ - do ideals of F [G ] contain ‘large enough’ relative augmentationideals?

Answer: No. H = 〈x , y |[x , y ] central〉 Heisenberg group.It was computed that resH(n) ∼ n6 (recall that γH(n) ∼ n4).However...

Be’eri Greenfeld (BIU) Residual girth of groups and algebras Spa 2019 6 / 11

Comparing groups and algebras

Consider group algebras: resG (n) vs. resF [G ](n)?Evidently: resF [G ](n) � resG (n) (normal subgroups give rise to ideals).

Question

Is resG (n) ∼ resF [G ](n)?

Motivation: Can n-Balls be more efficiently detected if we usering-theoretic homomorphic images? ‘Quantitative ideal congruenceproperty’ - do ideals of F [G ] contain ‘large enough’ relative augmentationideals?Answer: No. H = 〈x , y |[x , y ] central〉 Heisenberg group.It was computed that resH(n) ∼ n6 (recall that γH(n) ∼ n4).However...

Be’eri Greenfeld (BIU) Residual girth of groups and algebras Spa 2019 6 / 11

Heisenberg group algebra

Theorem (B.G.)

We have: resQ[H](n) � n4 log3(n).

Note: γH(n) = γF [H](n) ∼ n4 and therefore n4 � resF [H](n).Proof idea: Map F [H] onto products of matrix algebras using suitablesymbol presentations as split csa.What about other fields?

Theorem (B.G.)

If F is a finite field, assuming GRH we have: resF [H](n) � n4 log3(n).

Probably not equivalent to GRH... How about large fields?

Theorem (B.G.)

If F ⊇ Qab then resF [H](n) � n6−ε for every ε > 0.

Be’eri Greenfeld (BIU) Residual girth of groups and algebras Spa 2019 7 / 11

Heisenberg group algebra

Theorem (B.G.)

We have: resQ[H](n) � n4 log3(n).

Note: γH(n) = γF [H](n) ∼ n4 and therefore n4 � resF [H](n).Proof idea: Map F [H] onto products of matrix algebras using suitablesymbol presentations as split csa.What about other fields?

Theorem (B.G.)

If F is a finite field, assuming GRH we have: resF [H](n) � n4 log3(n).

Probably not equivalent to GRH... How about large fields?

Theorem (B.G.)

If F ⊇ Qab then resF [H](n) � n6−ε for every ε > 0.

Be’eri Greenfeld (BIU) Residual girth of groups and algebras Spa 2019 7 / 11

Heisenberg group algebra

Theorem (B.G.)

We have: resQ[H](n) � n4 log3(n).

Note: γH(n) = γF [H](n) ∼ n4 and therefore n4 � resF [H](n).Proof idea: Map F [H] onto products of matrix algebras using suitablesymbol presentations as split csa.What about other fields?

Theorem (B.G.)

If F is a finite field, assuming GRH we have: resF [H](n) � n4 log3(n).

Probably not equivalent to GRH... How about large fields?

Theorem (B.G.)

If F ⊇ Qab then resF [H](n) � n6−ε for every ε > 0.

Be’eri Greenfeld (BIU) Residual girth of groups and algebras Spa 2019 7 / 11

Comparing residual girth of algebras with wordgrowth/GK-dimension

For which algebras the residual girth growth coincides with the usualgrowth? (Evident examples: polynomial rings, graded algebras...)

Theorem (B.G.)

Let R be a finitely generated prime PI algebra. Then:resR(n) ∼ γR(n) ∼ nGKdim(R).

(Proof idea: Pass to a central localization which is finite and free over acentral subring, and use Hironaka decomposition to obtain a free moduleover a polynomial subring - and pull residual data along the freeextensions.)

Be’eri Greenfeld (BIU) Residual girth of groups and algebras Spa 2019 8 / 11

Comparing residual girth of algebras with wordgrowth/GK-dimension

For which algebras the residual girth growth coincides with the usualgrowth? (Evident examples: polynomial rings, graded algebras...)

Theorem (B.G.)

Let R be a finitely generated prime PI algebra. Then:resR(n) ∼ γR(n) ∼ nGKdim(R).

(Proof idea: Pass to a central localization which is finite and free over acentral subring, and use Hironaka decomposition to obtain a free moduleover a polynomial subring - and pull residual data along the freeextensions.)

Be’eri Greenfeld (BIU) Residual girth of groups and algebras Spa 2019 8 / 11

Comparing residual girth of algebras with wordgrowth/GK-dimension

In general, how big can be the gap between usual growth and residualgirth growth?

Theorem (B.G.)

Let f : N→ N be arbitrary.Then there exists a finitely generated, residually finite dimensional algebraR over an arbitrary field such that:

resR(n) > f (n) infinitely often;

GKdim(R) = 2.

As in groups, residual girth is related to algorithmic properties:

Proposition (B.G)

If R is a finitely presented residually finite ring (e.g. Fp[G ] for finitelypresented G ) then resR(n) is computable.

Be’eri Greenfeld (BIU) Residual girth of groups and algebras Spa 2019 9 / 11

Comparing residual girth of algebras with wordgrowth/GK-dimension

In general, how big can be the gap between usual growth and residualgirth growth?

Theorem (B.G.)

Let f : N→ N be arbitrary.Then there exists a finitely generated, residually finite dimensional algebraR over an arbitrary field such that:

resR(n) > f (n) infinitely often;

GKdim(R) = 2.

As in groups, residual girth is related to algorithmic properties:

Proposition (B.G)

If R is a finitely presented residually finite ring (e.g. Fp[G ] for finitelypresented G ) then resR(n) is computable.

Be’eri Greenfeld (BIU) Residual girth of groups and algebras Spa 2019 9 / 11

Comparing residual girth of algebras with wordgrowth/GK-dimension

In general, how big can be the gap between usual growth and residualgirth growth?

Theorem (B.G.)

Let f : N→ N be arbitrary.Then there exists a finitely generated, residually finite dimensional algebraR over an arbitrary field such that:

resR(n) > f (n) infinitely often;

GKdim(R) = 2.

As in groups, residual girth is related to algorithmic properties:

Proposition (B.G)

If R is a finitely presented residually finite ring (e.g. Fp[G ] for finitelypresented G ) then resR(n) is computable.

Be’eri Greenfeld (BIU) Residual girth of groups and algebras Spa 2019 9 / 11

Further research

What is the precise growth rate of resF [G ](n) for G nilpotent?

Is the residual girth of Grigorchuk’s group algebra exponential orintermediate? What about other branch group algebras?

What ring theoretic properties are reflected in the growth of theresidual girth?

Be’eri Greenfeld (BIU) Residual girth of groups and algebras Spa 2019 10 / 11

Thank You!

Questions?

Be’eri Greenfeld (BIU) Residual girth of groups and algebras Spa 2019 11 / 11