The structures of higher rank lattice actions on dendrites

Abstract

Let $\Gamma$ be a higher rank lattice acting on a nondegenerate dendrite $X$ with no infinite order points. We show that there exists a nondegenerate subdendrite $Y$ which is $\Gamma$-invariant and satisfies the following items: (1) There is an inverse system of finite actions $\{(Y_i,\Gamma ):i=1,2,3,\cdots \}$ with monotone bonding maps ${\varphi }_i:Y_{i+1}\to Y_i$ and with each $Y_i$ being a dendrite, such that $(Y,\Gamma |Y)$ is topologically conjugate to the inverse limit $(\underset{⟵}{\lim }(Y_i,\Gamma ),\Gamma )$. (2) The first point map $r:X\to Y$ is a factor map from $(X,\Gamma )$ to $(Y,\Gamma |Y)$; if $x\in X\setminus Y$, then $r(x)$ is an end point of $Y$ with infinite orbit; for each $y\in Y$, $r^{-1}(y)$ is contractible, that is there is a sequence $g_i\in \Gamma$ with $\mathrm{diam}(g_i{r}^{-1}(y))\to 0$.