Çankaya University Mathematics Department Seminars

JORGENSEN'S INEQULITY AND PURELY LOXODROMIC 2-GENERATOR FREE KLEINIAN GROUPS
İlker Yüce
-, Turkey
Özet : Let $\xi$ and $\eta$ be two non--commuting isometries of the hyperbolic $3$--space $\mathbb{H}^3$ such that $\langle\xi,\eta\rangle$ is a purely loxodromic free Kleinian group. Suppose that the inequality $$ \max\{d_{\gamma}z_0,d_{\beta\gamma\beta^{-1}}z_0\}\geq\max\{d_{\psi\phi\psi^{-1}}z_0\ :\ \psi,\ \phi\in\Psi_r=\{\xi,\eta^{-1},\eta,\xi^{-1}\}\} $$ holds for some $\gamma\in\Psi_r$ and $\beta\in\Psi_r-\{\gamma,\gamma^{-1}\}$ for $z_0\in\mathbb{H}^3$ the mid--point of the shortest geodesic segment joining the axes of $\gamma$ and $\beta\gamma\beta^{-1}$, where $d_{\gamma}z_0$ denotes the distance between $z_0$ and $\gamma z_0$. Let $A$ and $B$ be the matrices in PSL($2,\mathbb{C}$) representing the isometries $\gamma$ and $\beta$, respectively. If ${tr}(\cdot)$ denotes the trace and $\alpha=24.8692...$ is the unique real root of the polynomial $21 x^4 - 496 x^3 - 654 x^2 + 24 x + 81$ greater than $9$, in this talk I will prove that $$ |{tr}^2(A)-4|+|{tr}(ABA^{-1}B^{-1})-2|\geq 2\sinh^2\left(\frac{1}{4}\log\alpha\right)= 1.5937.... $$ Also a generalisation to finitely generated purely loxodromic free Klenian groups will be conjectured if time permits.
  Tarih : 04.03.2016
  Saat : 13:20
  Yer : R-213
  Dil : English
  Ek Dosya : Özet
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