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 $\phi_i: Y_{i+1}\rightarrow Y_i$ and with each $Y_i$ being a dendrite, such that $(Y, \Gamma|Y)$ is topologically conjugate to the inverse limit $(\underset{\longleftarrow}{\lim}(Y_i, \Gamma), \Gamma)$. (2) The first point map $r:X\rightarrow Y$ 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 ${\rm diam}(g_ir^{-1}(y))\rightarrow 0$.