Provable algorithm:

Currently, we expand intervals in small \(\epsilon\) steps and check for containment along the way. However, it is possible that on any given failed trial, our choosen value for \(\epsilon\) was too large and a smaller choice would be able to identify successful Ping Pong intervals. Then, we want to implement an algorithm that will test for Ping Pong intervals to arbritary precision, giving a sure answer for whether a given set of generators generate a free group or not. This algorithm will either output successful Ping Pong intervals or run indefinetely, decreasing the \(\epsilon\) step and testing for intervals as time goes on.

Figure 1

- Automatic Structures (some groups have faithful actions but are not free, how can we modify this algorithm to work for different groups?)

  - containment is based on a graph depending on the group (used to be consistent for free groups)

  - initial intervals are based on SVD instead of eigenvectors