# The computability of a fgoagog (intro)

I’m going to write up a result I think I’ve proved, which follows on from Gatterdam’s thesis. He showed that if a group is

The tactic will be to follow Gatterdam’s method for amalgamated products – if we can find an algorithm which solves the word problem of the fgoagog (decides if a word on the generators is equivalent to the identity) in

I will define an *admissible form* for words from the fgoagog, which is an alternating series of edge letters and words from the vertex groups, such that the edge letters make up a cycle starting and ending at the base vertex. Once we have a word in admissible form, we can reduce it by swapping the image of an edge group in one vertex with its counterpart in the vertex at the other end, and eventually a trivial word will end up as just a word on the base vertex, which is an

Clearly the process of putting a word in admissible form involves knowing something about the shape of the graph, so we need to make some