Primitive and measure-preserving systems of elements on the varieties
of metabelian and metabelian profinite groups

E. Timoshenko

Consider an ordered set (system) of elements \(\{ v_1,\ldots,v_l \}\), \(1 \leq l \leq r\), in the free group \(F_r\) of rank \(r\). Let \(G\) be a finite group. Define the verbal mapping \(\varphi_{\{ v_1,\ldots,v_l \} }\) from \(G^r\) into \(G^l\) by assigning to each \(\overline{g}= (g_1,\ldots, g_n) \in G^r\) the element \(( v_1(g_1,\ldots, g_r),\ldots,v_l(g_1,\ldots, g_r))\) in \(G^l.\) A system of elements \(\{ v_1,\ldots,v_l \}\) preserves measure on \(G\) if every \(\overline{g} \in G^l\) appears as an image under \(\varphi_{\{ v_1,\ldots,v_l \} }\) with probability \(|G|^{-l}.\) A system of elements \(\{ v_1,\ldots,v_l \}\), \(1 \leq l \leq r\), that preserves measure on every finite group \(G\) is called measure-preserving.

A system of elements \(\{v_1,\ldots,v_l \}\), \(1 \leq l \leq r\), in \(F_r\) is called primitive if it can be complemented to a basis of \(F_r.\)

The following conjecture by several authors about the connection between primitive elements and measure-preserving elements was formulated:

Conjecture 1. A system of elements \(\{v_1,\ldots, v_l\}\), \(1\leq l \leq r\), in a free group \(F_r\) is primitive if and only if the system preserves measure.

The conjecture was confirmed for \(l\geq r-1\) and, later, for \(l=1\).

Suppose that we consider as the finite groups \(G\) only the groups in some variety \(\mathfrak M\). Let \(V= V(\mathfrak M)\) be the verbal subgroup in \(F_r\) corresponding to this variety. For \(\{v_1,\ldots,v_l\}\) as in the definition of systems of measure-preserving elements, we may consider elements in the free group \(F_r(\mathfrak M )= F_r/V\) of \(\mathfrak M\).

Replacing in the definition of systems of elements an arbitrary finite group \(G\) with an arbitrary finite group in \(\mathfrak M\), and a free group \(F_r\), with a relatively free group \(F_r(\mathfrak M)\), we arrive at the notation of systems of elements in \(F_r(\mathfrak M)\) that preserve measure on \(\mathfrak M\).

By analogy with the definition of system of primitive elements in \(F_r\), we can define primitive systems of elements in \(F_r(\mathfrak M)\) as the systems that can be included in some basis of \(F_r(\mathfrak M)\).

The above-formulated conjecture can be expressed for the group variety \(\mathfrak M\):

Conjecture 2. A system of elements \(\{v_1,\ldots, v_l\}\), \(1\leq l \leq r\), in the free group \(F_r(\mathfrak M)\) is primitive if and only if the system preserves measure on \(\mathfrak M\).

We use the primitivity criteria for varieties of metabelian groups to prove the following theorems.

Theorem 1. Let \(S\) be a free metabelian group of rank \(r \geq 2\). An element \(v\) preserves measure on the variety \(\mathfrak A^2\) of all metabelian groups if and only if \(v\) is primitive.

Theorem 2. Let \(S\) be a free metabelian group of rank \(r \geq 2\). A system of elements \(\{v_1,\ldots, v_r\}\) preserve measure on the variety \(\mathfrak A^2\) of all metabelian groups if and only if they form a basis for \(S\).

Theorem 3. A system of elements \(\{v_1,\ldots, v_l\}, 1 \leq l \leq r,\) in a free profinite \(\mathfrak A^2\) - group \(\widehat{S}_r\) is primitive if and only if this system preserves measure on the variety of profinite \(\mathfrak A^2\) - groups.

Theorem 4. Suppose that \(v\) belongs to a free metabelian group \(S_r\) and \(\widehat{S}_r\) is the profinite completion of \(S_r\). The element \(v\) is primitive in \(S_r\) if and only if \(v\) is primitive in \(\widehat{S}_r\).

Theorem 5. Let elements \(\{v_1, \ldots, v_r\}\) be chosen in a free metabelian group \(S_r\). They constitute a basis for \(S_r\) if and only if they are a basis for the profinite completion \(\widehat{S}_r\) of \(S_r\).

See also the author's pdf version: