Abstract:We give an explicit construction of two $2$-generated subgroups $H,K\leq \SL(3,\Z)$ whose intersection is not finitely generated. The construction takes place inside the standard parabolic subgroup $\Z^2\rtimes \SL(2,\Z)\leq \SL(3,\Z)$. The main point is to identify $H\cap K$ with the stabilizer of a point for an affine action of a free group on $\Z^2$, and then to prove, using the Schreier graph of this action, that this stabilizer is not finitely generated. Furthermore, we prove that there exists a sequence of subgroups $H_q, K_q \leq \SL(3,\mathbb{Z})$ such that $\rank(H_q)=\rank(K_q)=4$, and \[ \rank(H_q\cap K_q)\geq q+1, \] while $H_q\cap K_q$ is finitely generated.
| Subjects: | Group Theory (math.GR); Geometric Topology (math.GT) |
| Cite as: | arXiv:2605.25080 [math.GR] |
| (or arXiv:2605.25080v1 [math.GR] for this version) | |
| https://doi.org/10.48550/arXiv.2605.25080 arXiv-issued DOI via DataCite (pending registration) |
From: Shengkui Ye [view email]
[v1]
Sun, 24 May 2026 13:45:25 UTC (9 KB)
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