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Resonances on geometrically finite graphs

Christian Arends, Carsten Peterson, Tobias Weich

Abstract

In analogy with the spectral theory of geometrically finite hyperbolic manifolds, we initiate the study of resonances on geometrically finite (q+1)-regular graphs of groups. We prove the meromorphic continuation of the resolvent of the adjacency operator on such spaces and give a geometric characterization of the resonant states. In contrast to the hyperbolic surfaces setting, geometrically finite graphs have only finitely many resonances and may be computed explicitly, yet exhibit many of the same qualitative phenomena as in the hyperbolic manifolds setting. Particularly interesting examples arise from algebraic curves over finite fields.

Resonances on geometrically finite graphs

Abstract

In analogy with the spectral theory of geometrically finite hyperbolic manifolds, we initiate the study of resonances on geometrically finite (q+1)-regular graphs of groups. We prove the meromorphic continuation of the resolvent of the adjacency operator on such spaces and give a geometric characterization of the resonant states. In contrast to the hyperbolic surfaces setting, geometrically finite graphs have only finitely many resonances and may be computed explicitly, yet exhibit many of the same qualitative phenomena as in the hyperbolic manifolds setting. Particularly interesting examples arise from algebraic curves over finite fields.

Paper Structure

This paper contains 31 sections, 28 theorems, 131 equations, 14 figures, 1 table.

Table of Contents

  1. Introduction
  2. Geometrically finite graphs of groups: definitions
  3. Graphs of groups
  4. Geometrically finite graphs of groups
  5. Adjacency operator on graphs of groups
  6. Geometrically finite graphs of groups: motivations
  7. Geometric finiteness in negative curvature
  8. Funnels and convex cocompact groups
  9. Cusps
  10. Arithmetic lattices
  11. Resolvent on model spaces
  12. Resolvent on a funnel
  13. Resolvent on the $(q+1)$-regular tree
  14. Convergence of the resolvent on weighted $\ell^2$-spaces
  15. Resolvent on a cusp
  16. ...and 16 more sections

Key Result

Theorem 1.1

Let $\mathcal{G}$ be an asymptotically $(q+1)$-regular graph of groups with vertex set $\mathcal{V}$. Then the resolvent is defined for $\lvert z(\mu) \rvert \geq \frac{\|A\|}{2 \sqrt{q}}$, holomorphic in that range, and admits a finitely meromorphic continuation as a family of operators for $\lvert \mu \rvert > q^{\frac{1}{2} - N}$, where $o$ denotes an arbitrary base point in the compact core

Figures (14)

  • Figure 2.1: The top image above shows a 3-regular funnel. The bottom image above shows a 3-regular orbifold funnel arising as the quotient of the funnel on the LHS by $\mathbb{Z}/2 \times \mathbb{Z}/2$. The first factor of $\mathbb{Z}/2$ swaps the two descendant branches of $o$. The second factor of $\mathbb{Z}/2$ swaps the two descendant branches of $u_1$ and the two descendant branches of $u_2$ (which are the two lifts of $u$ under the quotient mapping). Only vertex groups are shown; each edge group is equal to the vertex group of the vertex of the edge further away from $o$. Each label $\mathbb{Z}/2$ on the bottom image really corresponds to the subgroup $1 \times \mathbb{Z}/2 < \mathbb{Z}/2 \times \mathbb{Z}/2$. Note that the bottom image is not 3-regular as a graph, but it is 3-regular as a graph of groups.
  • Figure 2.2: An example of a 3-regular cusp. See also Figure \ref{['fig_nagao']}.
  • Figure 2.3: An example of a 3-regular geometrically finite graph with compact core (in pink), two funnels (in blue), and two cusps (in green).
  • Figure 3.1: The LHS shows the hyperbolic cylinder corresponding to an element $\gamma$ with translation length 4 acting on the 4-regular tree. The RHS shows the parabolic cylinder for the Bruhat-Tits tree of $\textnormal{SL}(2, \mathbb{F}_2((T)))$.
  • Figure 3.2: Example of the Iwasawa folding for the Bruhat-Tits tree of $\textnormal{PGL}(2, \mathbb{F}_2((T)))$. The red vertical line represents the central geodesic that we are folding onto "from the perspective of $-\infty$". Each brown horizontal line represents points on the same horocycle. The point $v_1$ has level 2 and branch point $-1$, so for example it might correspond to $T^2T^{-1}01 K$. The point $v_2$ has level 2 and branch point $-2$, so for example it might correspond to $T^2T^{-2}01 K$.
  • ...and 9 more figures

Theorems & Definitions (58)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Theorem 3.1: Paulin paulin, Théorème 1.1
  • Proposition 3.2
  • ...and 48 more