The structures of higher rank lattice actions on dendrites

TL;DR

This paper studies how higher rank lattices act on dendrites, showing the existence of invariant subdendrites with inverse system structures and describing the nature of the factor map from the original dendrite.

Contribution

It introduces a new structure theorem for higher rank lattice actions on dendrites, revealing inverse system decompositions and properties of the factor map.

Findings

Existence of a nondegenerate invariant subdendrite with inverse system structure.

The factor map from the original dendrite to the invariant subdendrite has specific properties.

Points outside the invariant subdendrite map to endpoints with infinite orbits.

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 ${\rm diam}(g_ir^{-1}(y))\rightarrow 0$.