TL;DR
This paper investigates the spectral dimension of spheres by associating growth graphs and length operators to homogeneous spaces, establishing bounds, and computing explicit spectral dimensions.
Contribution
It introduces a method linking growth graphs and length operators to determine spectral dimensions of spheres, advancing understanding of geometric analysis on homogeneous spaces.
Findings
Spectral dimension of spheres is computed explicitly.
Spectral dimension bounds are established via length operator summability.
Method links growth graph properties to spectral analysis.
Abstract
In this paper, we associate a growth graph and a length operator to a quotient space of a semisimple compact Lie group. Under certain assumptions, we show that the spectral dimension of a homogeneous space is greater than or equal to summability of the length operator. Using this, we compute spectral dimensions of spheres.
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