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What are some intuitive ways to think about one-relator groups?

I am aware of the Freiheitsatz, and Bass-Serre theory. What I'm interested in are ways people who work extensively with one-relator groups think about them. I'd like some help with the intuition.

To elaborate a bit more: Are there diagrammatic tools to study these? What are the go-to rewriting procedures? What is an intuitive description of the realization of these as HNN extensions (as found in Lyndon and Schupp, for example). What are the go-to techniques and tools in the modern viewpoint? Although I'm aware of the Freiheitsatz and Bass-Serre theory, an expert description of how these are employed to study one-relator groups wouldn't be off topic.

Thanks in advance!

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There are combinatorial methods such as Dehn diagrams and pictures which are used in proving results such as non-triviality. It involves arguments using curvature, euler characteristics etc. I hope someone who knows about these can help. – Hesky Cee Feb 17 '14 at 20:42

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