The conjecture is not true for arbitrary group words ( w ). A counterexample can be constructed using a specific group word and analyzing the density of tuples satisfying the identity.

Consider the group word ( w(a,b,c) = [a,b]c ), where ( [a,b] = aba^{-1}b^{-1} ). The variety ( \mathfrak{U}_w ) consists of all groups ( G ) where ( [a,b]c = e ) for all ( a,b,c \in G ). This implies...

Rephrase. For a fixed group word w in n variables and a finite group G, write p_w(G) = |{(a1,…,an) in G^n : w(a1,…,an)=1}| / |G|^n. Your question asks whether for every word w there is a constant ε(w)>0 such that, for all finite groups G, p_w(G) > 1 − ε(w) implies that G satisfies the identity w≡1 (i.e. G lies in the variety U_w defined by w).

What is known.

• Commutator word. For w=[x,y] (abeli...

This is a fascinating question about generalizing the Erdős-Turán theorem to arbitrary group words. Let me analyze this carefully.

The conjecture asks whether for every group word , there exists a positive constant such that a finite group satisfies the identity if and only if the proportion of tuples satisfying exceeds $1 - \epsilon...

Generalization of Erdős-Turán's 5/8 Theorem

This is an interesting question about generalizing the Erdős-Turán theorem to arbitrary group words. Let me analyze it carefully.

Understanding the Question

The question asks whether for any group word over variables, there exists a positive constant such that a finite group belongs to the variety (where...

The problem you raised is exactly the problem of deciding whether a given word w is a probabilistic identity.
For a word w in the free group F = 〈x1 , … , xn 〉 write

P G (w) = |{(a1 , … , an ) ∈ G n : w(a1 , … , an ) = 1}| ⁄ |G| n

(the probability that a random n–tuple in the finite group G satisfies w = 1). The question is:

(∗) Does every non-trivial word w admit ε(w) > 0 such that
...

First, let us restate the question in the language of “probabilistic identities.” Fix a word
  w=w(x₁,…,xₙ)∈Fₙ,
and let 𝒰_w be the variety of (all) groups satisfying the identity w=1 . For a finite group G we set

  p_G(w) = |{(g₁,…,gₙ)∈Gⁿ : w(g₁,…,gₙ)=1}| ÷ |G|ⁿ.

The conjecture is that for every word w there is an ε(w)>0 so that

  G∈𝒰_w ⟺ p_G(w)>1−ε(w).

Equivalently o...

<think> I need to understand this question. It's about generalizing the Erdős-Turán theorem, which says that for the commutator word [a,b], if the probability that a random pair commutes is greater than 5/8, then the group is abelian. Similarly, for higher commutators, there's a threshold for nilpotency.

The question is whether for any group word w, there's some epsilon(w) > 0 such that for any f...

The problem you ask for is usually formulated as follows.

For a word w in n free generators put

  p w (G)= |{(g1 ,…,gn )∈G n :w(g1 ,…,gn )=1}| /|G| n .

Question (P. M. Neumann, Kourovka Notebook, Problem 11.36).
Does every non–trivial word w admit a constant ε(w)>0 such that, for every finite group G,

  p w (G)>1−ε(w) ⇒ w is an identity of G ?

Equivalently, is

  sup {p w (G) : G f...

This is an excellent and very natural question that gets to the heart of a field known as probabilistic group theory. The short answer to your question is no, the conjecture is not true for an arbitrary group word .

The property you are asking about has a name. A variety of groups (defined by a set of laws, or in your case, a single law ) is called **probabilistical...