AI 中文总结
研究高阶连通半单实代数群离散子群增长指标函数，证明非初等相对博雷尔阿诺索夫群的该函数在非共线方向严格凹，通过建立曼哈顿超曲面$\mathcal C^1$光滑性及证明非初等$\theta$ - 横向群相关性质得出结论。
AI 中文摘要
设$\Gamma$为高阶连通半单实代数群的离散子群。增长指标函数$\psi_\Gamma$记录$\Gamma$元素在正外尔腔$\mathfrak a^+$中嘉当投影的方向指数增长。我们证明若$\Gamma$是非初等相对博雷尔阿诺索夫群，则$\psi_\Gamma$在非共线方向上严格凹。通过建立曼哈顿超曲面的$\mathcal C^1$光滑性来证明，该超曲面定义为临界指数映射$\phi\mapsto\delta^\phi(\Gamma)$的单位水平集。更一般地，对于非初等$\theta$ - 横向群，我们证明在$\theta$ - 曼哈顿超曲面上每一点附近的局部$\mathcal C^1$正则性，该超曲面在$\theta$ - 极限锥上为正且在无穷远处有临界间隙。特别地，相对$\theta$ - 阿诺索夫群的$\theta$ - 曼哈顿超曲面是全局$\mathcal C^1$的，且其$\theta$ - 增长指标函数在非共线方向上严格凹。
英文摘要
Let $Γ$ be a discrete subgroup of a connected semisimple real algebraic group of higher rank. The growth indicator function $ψ_Γ$ records the directional exponential growth of the Cartan projections of elements of $Γ$ in the positive Weyl chamber $\mathfrak a^+$. We prove that if $Γ$ is a non-elementary relatively Borel Anosov group, then $ψ_Γ$ is strictly concave on non-collinear directions. We prove this by establishing the $\mathcal C^1$-smoothness of the Manhattan hypersurface, defined as the unit level set of the critical-exponent map $ϕ\mapstoδ^ϕ(Γ)$. More generally, for a non-elementary $θ$-transverse group, we prove local $\mathcal C^1$-regularity near every point of the $θ$-Manhattan hypersurface that is positive on the $θ$-limit cone and has a critical gap at infinity. In particular, the $θ$-Manhattan hypersurface is globally $\mathcal C^1$ for relatively $θ$-Anosov groups, and their $θ$-growth indicator functions are strictly concave on non-collinear directions.
Comments31 pages. Comments welcome!
