Free subgroups vs law  Consider the following two conditions for a group $G$:
(1) $G$ does not satisfy a nontrivial law.
(2) $G$ contains a non-abelian free subgroup. 
Obviously (2) implies (1) and it is easy to construct torsion groups that do not satisfy any law (e.g., the direct product of all finite groups).  Thus (1) does not imply (2) in general. However the following question seems open:

Are (1) and (2) equivalent for profinite groups?

Here is a similar question:

Suppose that a residually finite group does not satisfy a law. Does its profinite completion contain a nonabelian free subgroup?

These questions may be thought of as possible generalizations of the Tits alternative to residually finite (or profinite) groups. 
The answer to the second question is positive for finitely generated p-groups. This follows from [Wilson, Zelmanov,  Identities for Lie algebras of pro-p groups, JPAA 81(192), 103-109], where the authors prove that if $G$ is a finitely generated residually finite p-group, then either it is finite (and hence satisfies a law), or its profinite completion contains a nonabelian free group. In the general case they only prove that if the profinite completion of a residually finite group does not contain a nonabelian free subgroup, then some Lie algebra associated to the group satisfies a law. 
 A: Here is a partial solution to the conjecture.  I don't know how strong it is, but I feel like posting it.
Every profinite group $G$ is a compact Hausdorff space, and therefore a Baire space, and so is $G \times G$.  This exactly means that the intersection of countably many open dense sets is still dense.  Now a word $w(a,b)$ in two generators $a$ and $b$ yields a function
$$w:G \times G \to G.$$
The word is satisfied when $w(a,b) = 1$.  It is a law if $w(a,b) = 1$ always, i.e., if $w$ sends all of $G \times G$ to 1.  In general $w$ is satisfied on some closed set of $G \times G$.  Suppose that every word is only satisfied on a nowhere dense set in $G \times G$.  Then the Baire category theorem says that there is a comeager set of pairs $(a,b)$ that don't satisfy any word, and there then is a free subgroup of rank 2.
On the other hand, suppose that a word $w$ is satisfied on a closed set $C$ that is somewhere dense.  Then because the topology on $G \times G$ is induced by homomorphisms to finite groups, $C$ contains the inverse image of some point under a homomorphism
$$(\alpha,\alpha):G \times G \to A \times A,$$
where $A$ is a finite group.  Suppose that $C$ contains $(\alpha^{-1}(1),\alpha^{-1}(1))$, and let $n$ be the exponent of $A$.  Then the word $w'(a,b) = w(a^n,b^n)$ is a law on $G$.
Where I get stuck is that $C$ might instead contain the inverse image of some different element of $A \times A$.  For instance, suppose that $G = C_2 \ltimes C_3^\infty$ is a semidirect product, where the $C_2$ acts (by conjugation) on each $C_3$ by inverting it.  Then the word $w(a) = a^2$ is satisfied on an open set that does not contain the identity.  Given such a word (in two generators), I tried to make another word that contains an open set around the identity, but I did not succeed.
A: Since Henry asked, here is a reference: Aner Shalev in the first chapter (Lie Methods in the Theory of pro-$p$ Groups) in New Horizons in pro-$p$ Groups posed 4 conjectures in decreasing order of strength:


*

*Let $G$ be a finite $p$-group satisfying some identity $w$ with probability $\epsilon>0$. Then $G$ satisfies some identity depending only on $w$, $p$, and $\epsilon$.

*Let $G$ be a finitely generated pro-$p$ group satisfying some identity with positive probability. Then $G$ satisfies some identity.

*Let $G$ be a finitely generated pro-$p$ group satisfying some coset-identity. Then $G$ satisfies some identity.

*Let $G$ be a $k$-generated pro-$p$ group satisfying some identity on all generating $k$-tuples. Then $G$ satisfies some identity.
There is a discussion there about what is known. I am not sure what developments occurred since the book appeared and what is known about the profinite case in general. I think it may have been mentioned by Shalev or Mann in other surveys. 
