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 groupsubgroup?

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.

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 an equivalent a similar question:

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

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 a residually finite group does not contain a nonabelian free subgroup, then some Lie algebra associated to the group satisfies a 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 an equivalent question:

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

Positive answer to the first question would imply positive answer to the second one.

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 a residually finite group does not contain a nonabelian free subgroup, then some Lie algebra associated to the group satisfies a 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 group?

Positive answer to the first question would imply positive answer to the second one. 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 p-torsion groupsfinitely 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 a residually finite group does not contain a nonabelian free subgroup, then some Lie algebra associated to the group satisfies a 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 group?

Positive answer to the first question would imply positive answer to the second one. 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 p-torsion groups. This follows from [Wilson, Zelmanov, Identities for Lie algebras of pro-p groups, JPAA 81(192), 103-109]. In the general case they only prove that if a residually finite group does not contain a nonabelian free subgroup, then some Lie algebra associated to the group satisfies a 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?

The answer

Here is positive for pro-p groups. Indeed it was proved by Zelmanov a similar question:

Suppose that a pro-p residually finite group satisfying does not satisfy a lawis finite. I Does its profinite completion contain a nonabelian free group?

Positive answer to the first question would imply positive answer to the second one. These questions may be surprised if (1) and thought of as possible generalizations of the Tits alternative to residually finite (2) were equivalent for all or profinite) groups. But I do

The answer to the second question is positive for p-torsion groups. This follows from [Wilson, Zelmanov, Identities for Lie algebras of pro-p groups, JPAA 81(192), 103-109]. In the general case they only prove that if a residually finite group does not know any counterexamplecontain a nonabelian free subgroup, then some Lie algebra associated to the group satisfies a 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?

The answer is positive for pro-p groups. Indeed it was proved by Zelmanov that a pro-p group satisfying a law is finite. I would be surprised if (1) and (2) were equivalent for all profinite groups. But I do not know any counterexample.