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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2507.22266 |
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| _version_ | 1866908472698732544 |
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| author | Belolipetsky, Mikhail Hurtado, Sebastian |
| author_facet | Belolipetsky, Mikhail Hurtado, Sebastian |
| contents | The height gap theorem states that the finite subsets $F$ of matrices generating non-virtually solvable groups have normalized height $\widehat{h}(F)$ bounded below by a constant. It was first proved by Breuillard and another proof was given later by Chen, Hurtado and Lee. In this paper we show that when the set $F$ is contained in a maximal arithmetic subgroup $Γ$ of $G = PGL_2(\mathbb{R})^a \times PGL_2(\mathbb{C})^b$, $a+b \ge 1$, the height bound for the case when $F$ generates a Zariski dense subgroup of $G$ over $\mathbb{R}$ is proportional to $\log(covol(Γ))$, the function of the covolume of $Γ$. This result strengthens the theorem for the lattices of large covolume and has various applications including a strong version of the arithmetic Margulis lemma for $PGL_2(\mathbb{R})^a \times PGL_2(\mathbb{C})^b$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_22266 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A strong height gap theorem for $PGL_2$ Belolipetsky, Mikhail Hurtado, Sebastian Group Theory 20G25, 11F06, 20H10, 22E40 The height gap theorem states that the finite subsets $F$ of matrices generating non-virtually solvable groups have normalized height $\widehat{h}(F)$ bounded below by a constant. It was first proved by Breuillard and another proof was given later by Chen, Hurtado and Lee. In this paper we show that when the set $F$ is contained in a maximal arithmetic subgroup $Γ$ of $G = PGL_2(\mathbb{R})^a \times PGL_2(\mathbb{C})^b$, $a+b \ge 1$, the height bound for the case when $F$ generates a Zariski dense subgroup of $G$ over $\mathbb{R}$ is proportional to $\log(covol(Γ))$, the function of the covolume of $Γ$. This result strengthens the theorem for the lattices of large covolume and has various applications including a strong version of the arithmetic Margulis lemma for $PGL_2(\mathbb{R})^a \times PGL_2(\mathbb{C})^b$. |
| title | A strong height gap theorem for $PGL_2$ |
| topic | Group Theory 20G25, 11F06, 20H10, 22E40 |
| url | https://arxiv.org/abs/2507.22266 |