It is shown that the subgroup membership problem for a virtually free group can be decided in polynomial time when all group elements are represented by so-called power words, i.e., words of the form \(p_1^{z_1} p_2^{z_2} \cdots p_k^{z_k}\). Here the \(p_i\) are explicit words over the generating set of the group and all \(z_i\) are binary encoded integers. As a corollary, it follows that the subgroup membership problem for the matrix group \(\textsf{GL}(2,\mathbb {Z})\) can be decided in polynomial time when elements of \(\textsf{GL}(2,\mathbb {Z})\) are represented by matrices with binary encoded integers. For the same input representation, it also shown that one can compute in polynomial time the index of a given finitely generated subgroup of \(\textsf{GL}(2,\mathbb {Z})\).

结果表明，当所有群元素都由所谓的幂词表示时，可以在多项式时间内确定虚拟自由群的子群成员资格问题，即形式为 \(p_1^{z_1} p_2^{z_2} 的词\cdots p_k^{z_k}\)。这里的\(p_i\)是组生成集上的显式单词，所有\(z_i\)都是二进制编码的整数。作为推论，当\( \textsf{GL}( 2,\mathbb {Z})\)由具有二进制编码整数的矩阵表示。对于相同的输入表示，它还表明可以在多项式时间内计算\(\textsf{GL}(2,\mathbb {Z})\) 的给定有限生成子群的索引。