Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2403.05089 |
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Inhaltsangabe:
- Let $\mathcal{T}$ be a locally finite tree whose geometric boundary has infinitely many points. Suppose that a non-amenable group $\G$ acts isometrically and geometrically on the tree $\mathcal{T}$. In this paper, we show that if the length spectrum is Diophantine, then there exists a continuous function $C$ on $\mathcal{T}^2$ such that the heat kernel $p(t,x,y)$ of $\mathcal{T}$ satisfies $$\lim_{t\rightarrow \infty}t^{3/2}e^{λ_0t}p(t,x,y)=C(x,y)$$ for any $x,y\in \mathcal{T}$. Here, $λ_0$ is the bottom of the spectrum of the Laplacian on $\mathcal{T}$.