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Results tagged with one-relator-groups
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user 1345
Finitely presented groups having a single relator
7
votes
Accepted
1-related groups
Check out the proof of Theorem 4.1 of Joe Masters' paper. Given a 1-relator group presentation, realize the free group as the fundamental group of a compact surface (with boundary), and the relator as …
- 68.9k
12
votes
Accepted
Freiheitssatz implies a finitely generated one relator group embeds in a two-generator one r...
Yes, this follows from the Freiheitssatz. Assume that the 1-relator group is defined by $G=\langle g_1,\ldots, g_n | R(g_1,\ldots,g_n)\rangle$, such that the relator $R(g_1,\ldots, g_n)$ is cyclically …
- 68.9k
4
votes
Free splittings of one-relator groups
Your question is reminiscent of Jaco's lemma. A special case of Jaco's lemma applies to a 2-handle attached to the boundary of a (3-dimensional) handlebody $H$ (which has free fundamental group). If t …
- 68.9k
18
votes
Accepted
Free splittings of one-relator groups
$\newcommand{\rank}{\operatorname{rank}}$I think Grushko plus the Freiheitssatz does the trick.
Suppose that $G=A \ast B$ is a one-relator group which splits as a free product non-trivially. By Grushk …
- 68.9k