Deficiency and abelianized deficiency of some virtually free groups
Let $Q_m$ be the HNN extension of $\Z/m \times \Z/m$ where the stable letter conjugates the first factor to the second. We explore small presentations of the groups $\Gamma_{m,n}=Q_m \ast Q_n$. We show that for certain choices of (m,n), for example (2,3), the group $\Gamma_{m,n}$ has a relation gap...
| Κύριοι συγγραφείς: | , |
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| Μορφή: | Journal article |
| Γλώσσα: | English |
| Έκδοση: |
2006
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| Περίληψη: | Let $Q_m$ be the HNN extension of $\Z/m \times \Z/m$ where the stable letter conjugates the first factor to the second. We explore small presentations of the groups $\Gamma_{m,n}=Q_m \ast Q_n$. We show that for certain choices of (m,n), for example (2,3), the group $\Gamma_{m,n}$ has a relation gap unless it admits a presentation with at most 3 defining relations, and we establish restrictions on the possible form of such a presentation. We then associate to each (m,n) a 3-complex with 16 cells. This 3-complex is a counterexample to the D(2) conjecture if $\Gamma_{m,n}$ has a relation gap. |
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