Combinatorial classification of finitely generated virtually free groups
For a finitely generated virtually free group \(G\) denote by \(m_G\) the least common multiple of the orders of the finite subgroups in \(G\) and let \(b_G(\lambda)=\) number of free subgroups of index \(\lambda m_G\) in \(G\). The author introduces an equivalence \(\sim_a\) by \[ A\sim_a B\Leftrightarrow m_A=m_B\quad\text{and}\quad b_A(\lambda)\sim b_B(\lambda),\quad\lambda\to\infty, \] provided that both \(A\) and \(B\) are infinite. He extends this definition to the class of all finitely generated virtually free groups by setting \(A\sim_a B\Leftrightarrow|A|=|B|\) if one of the groups involved is finite. The main result of the present paper is a characterization this equivalence in terms of structural invariants. In particular, this characterization implies that a \(\sim_a\)-equivalence class of finitely generated virtually free groups contains only a finite number of nonisomorphic groups.
finitely generated virtually free groups
scientific article; zbMATH DE number 1076275
scientific article