Tent property and directional limit sets for selfjoinings of hyperbolic manifolds
Abstract
Let $\Gamma=(\rho_1\times \rho_2)(\Delta)$ where $\rho_1,\rho_2:\Delta\to \operatorname{SO}^\circ(n,1)$ are convex cocompact representations of a finitely generated group $\Delta$. (1): We provide a sharp pointwise bound on the growth indicator function $\psi_\Gamma$ by a tent function: for any $v=(v_1, v_2)\in {\mathbb R}^2$: $$ \psi_\Gamma(v)\le \min (v_1 \operatorname{dim}_{\operatorname{H}} \Lambda_{\rho_1}, v_2 \operatorname{dim}_{\operatorname{H}} \Lambda_{\rho_2}).$$ We obtain several interesting consequences including the gap and rigidity property on the critical exponent. (2): Generalizing this, we propose a conjecture that $\psi_\Gamma$ is at most the half sum of all positive roots for any Anosov subgroup of a semisimple real algebgraic group of rank at least $2$. We confirm this conjecture for Zariski dense Anosov subgroups of Hitchin groups. (3): For each $v$ in the interior of the limit cone of $\Gamma$, we prove the following on the $v$directional conical limit set $\Lambda_v\subset \mathbb S^{n1}\times \mathbb S^{n1}$: $$ \tfrac{\psi_\Gamma(v)}{\max{(v_1, v_2)} }\le \operatorname{dim}_H \Lambda_v \le \tfrac{\psi_\Gamma(v)}{\min{(v_1, v_2)} }.$$ We also study the local behavior of the higher rank PattersonSullivan measures on each $\Lambda_v$.
 Publication:

arXiv eprints
 Pub Date:
 December 2021
 arXiv:
 arXiv:2112.00877
 Bibcode:
 2021arXiv211200877K
 Keywords:

 Mathematics  Geometric Topology;
 Mathematics  Dynamical Systems
 EPrint:
 31 pages, 6 figures, A conjecture on the size of the growth indicator function added