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We give a simple algorithm to solve the subgroup membership problem forvirtually free groups. For a fixed virtually free group with a fixed generatingset $X$, the subgroup membership problem is uniformly solvable in time$O(n\log^*(n))$ where $n$ is the sum of the word lengths of the inputs withrespect to $X$. For practical purposes, this can be considered to be lineartime. The algorithm itself is simple and concrete examples are given to showhow it can be used for computations in $\mathrm{SL}(2,\mathbb Z)$ and$\mathrm{GL}(2,\mathbb Z)$. We also give an algorithm to decide whether afinitely generated subgroup is isomorphic to a free group.

A fast algorithm for Stallings foldings over virtually free groups