Bounded projections to the $\mathcal{Z}$-factor graph
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866913814228762624 |
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| author | Clay, Matt Uyanik, Caglar |
| author_facet | Clay, Matt Uyanik, Caglar |
| contents | Suppose $G$ is a free product $G = A_1 * A_2* \cdots * A_k * F_N$, where each of the groups $A_i$ is torsion-free and $F_N$ is a free group of rank $N$. Let $\mathcal{O}$ be the deformation space associated to this free product decomposition. We show that the diameter of the projection of the subset of $\mathcal{O}$ where a given element has bounded length to the $\mathcal{Z}$-factor graph is bounded, where the diameter bound depends only on the length bound. This relies on an analysis of the boundary of $G$ as a hyperbolic group relative to the collection of subgroups $A_i$ together with a given non-peripheral cyclic subgroup. The main theorem is new even in the case that $G = F_N$, in which case $\mathcal{O}$ is the Culler-Vogtmann outer space. In a future paper, we will apply this theorem to study the geometry of free group extensions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_17664 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Bounded projections to the $\mathcal{Z}$-factor graph Clay, Matt Uyanik, Caglar Group Theory Geometric Topology 20F65, 20E08 Suppose $G$ is a free product $G = A_1 * A_2* \cdots * A_k * F_N$, where each of the groups $A_i$ is torsion-free and $F_N$ is a free group of rank $N$. Let $\mathcal{O}$ be the deformation space associated to this free product decomposition. We show that the diameter of the projection of the subset of $\mathcal{O}$ where a given element has bounded length to the $\mathcal{Z}$-factor graph is bounded, where the diameter bound depends only on the length bound. This relies on an analysis of the boundary of $G$ as a hyperbolic group relative to the collection of subgroups $A_i$ together with a given non-peripheral cyclic subgroup. The main theorem is new even in the case that $G = F_N$, in which case $\mathcal{O}$ is the Culler-Vogtmann outer space. In a future paper, we will apply this theorem to study the geometry of free group extensions. |
| title | Bounded projections to the $\mathcal{Z}$-factor graph |
| topic | Group Theory Geometric Topology 20F65, 20E08 |
| url | https://arxiv.org/abs/2306.17664 |