[Paper Review] The first $L^p$-cohomology of some finitely generated groups and $p$-harmonic functions

Michael J. Puls|ArXiv.org|Sep 7, 2005

Geometry and complex manifolds4 references4 citations

TL;DR

This paper establishes a connection between the first $L^p$-cohomology of finitely generated groups and $p$-harmonic functions, showing that for groups with polynomial growth of degree $d > p \geq 2$, nonzero cohomology classes can be represented by functions in $L^{\frac{pd}{d-p}}(G)$. It further proves that for nonamenable groups, $H^1(G, L^p(G)) \subseteq H^1(G, L^{p'}(G))$ when $p' \geq p$, highlighting a key distinction from amenable groups.

ABSTRACT

Let $G$ be a finitely generated infinite group and let $p > 1$. In this paper we make a connection between the first $L^p$-cohomology space of $G$ and $p$-harmonic functions on $G$. We also describe the elements in the first $L^p$-cohomology space of groups with polynomial growth, and we give an inclusion result for nonamenable groups.

Motivation & Objective

• To establish a link between the first $L^p$-cohomology of finitely generated groups and $p$-harmonic functions.
• To characterize elements in the first $L^p$-cohomology space for groups with polynomial growth.
• To investigate inclusion relations between $H^1(G, L^p(G))$ and $H^1(G, L^{p'}(G))$ for nonamenable groups.
• To clarify the role of $p$-harmonic functions in the structure of $L^p$-cohomology spaces.

Proposed method

• Uses the group ring $\mathbb{R}G$ and convolution to define the space $D^p(G)$ of functions with $p$-integrable differences under right translations.
• Introduces a norm on $D^p(G)$ based on the $L^p$-norms of $f*(s-1)$ for $s$ in a symmetric generating set $S$, and quotients out by constants to form $D^p(G)/\mathbb{R}$.
• Applies the theory of reduced cohomology $\overline{H}^1_{(p)}(G)$ and decomposes $D^p(G)$ into $\overline{\mathbb{R}G}_{D^p(G)}$ and $HD^p(G)$, the space of harmonic functions.
• Leverages the equivalence of condition $(IS)_d$ and $S_d$ for groups of polynomial growth to relate cohomology to integrability in $L^r$-spaces.
• Uses the fact that $HD^p(G) = \mathbb{R}$ for groups with polynomial growth to isolate nontrivial cohomology classes.
• Applies approximation arguments in $D^p(G)$ and $L^r(G)$ to show that limits of sequences in $\mathbb{R}G$ converge in $L^{\frac{pd}{d-p}}(G)$, establishing representation in higher integrability classes.

Experimental results

Research questions

• RQ1 How are $p$-harmonic functions related to the first $L^p$-cohomology of a finitely generated group?
• RQ2 For groups with polynomial growth of degree $d$, what $L^r$-space contains representatives of nonzero classes in $H^1(G, L^p(G))$?
• RQ3 Does $H^1(G, L^p(G))$ embed into $H^1(G, L^{p'}(G))$ for nonamenable groups when $p' > p$?
• RQ4 Can the inclusion $H^1(G, L^p(G)) \subseteq H^1(G, L^{p'}(G))$ fail for amenable groups?

Key findings

• For a finitely generated group $G$ with polynomial growth of degree $d > p \geq 2$, every nonzero class in $H^1(G, L^p(G))$ can be represented by a function in $L^{\frac{pd}{d-p}}(G)$, as shown in Theorem 5.4.
• The space $HD^p(G)$ of $p$-harmonic functions on $G$ is equal to $\mathbb{R}$ for groups with polynomial growth, implying that nontrivial cohomology classes arise from non-constant functions in $D^p(G)$.
• For nonamenable groups, $H^1(G, L^p(G)) \subseteq H^1(G, L^{p'}(G))$ holds for all $p' \geq p$, as established in Proposition 6.2.
• The inclusion fails for amenable groups: there exist functions in $D^p(\mathbb{Z})$ that represent nonzero classes in $H^1(\mathbb{Z}, L^p(\mathbb{Z}))$ but lie in the zero class of $H^1(\mathbb{Z}, L^{p'}(\mathbb{Z}))$ for $p' > p$, as demonstrated by the example in Section 6.
• The norm on $D^p(G)/\mathbb{R}$ is equivalent to $\|f\|_{D(p)} = \left(\sum_{s \in S} \|f*(s-1)\|_p^p\right)^{1/p}$, which is independent of the choice of symmetric generating set $S$.
• The closure $\overline{\mathbb{R}G}_{D^p(G)}$ is a proper subspace of $D^p(G)$ when $G$ has polynomial growth of degree $d > p$, which implies that nontrivial cohomology classes cannot be represented by $L^p$-functions.