[Paper Review] l^p-cohomology for groups of type FP_n

Elias Kappos|ArXiv.org|Oct 31, 2005

Algebraic Geometry and Number Theory6 references3 citations

TL;DR

This paper establishes duality between reduced $l^p$-homology and $l^q$-cohomology for groups of type $FP_n$ with $\frac{1}{p} + \frac{1}{q} = 1$, and proves the vanishing of reduced $l^p$-homology and cohomology for such groups under conditions like having a central element of infinite order, infinitely many finite conjugacy classes, being nilpotent, or having polynomial growth. The key contribution is a general vanishing result for $l^p$-cohomology in the context of $FP_n$ groups with structural constraints on the group's center and conjugacy classes.

ABSTRACT

Let G be a group of type FP_n and let p>1. In this paper we show that the reduced l^p-homology of G is dual to the reduced l^q-cohomology for \frac{1}{p}+\frac{1}{q}=1. In our main theorem we show that for a group of type FP_n with a central element of infinite order the reduced l^p-cohomology vanishes. We generalize this fact for groups with infinitely many elements in the center of the group, for groups which are FCC, for groups with infinitely many finite conjugacy classes, for nilpotent groups, and for groups of polynomial growth.

Motivation & Objective

• To establish duality between reduced $l^p$-homology and $l^q$-cohomology for groups of type $FP_n$ with $\frac{1}{p} + \frac{1}{q} = 1$.
• To prove the vanishing of reduced $l^p$-homology and cohomology for $FP_n$ groups under structural conditions such as having a central element of infinite order.
• To generalize the vanishing result to groups with infinitely many finite conjugacy classes, nilpotent groups, and groups of polynomial growth.
• To extend the duality and vanishing theorems beyond the $l^2$-case to $l^p$-cohomology for $p > 1$ using functional analytic and group cohomological techniques.

Proposed method

• Use of projective resolutions of $\mathbb{Z}$ over $\mathbb{Z}[G]$ to define $l^p$-homology and cohomology via tensor and Hom constructions with $l^p(G)$.
• Definition of reduced $l^p$-(co)homology by taking the closure of images in the norm topology, ensuring completeness and duality.
• Application of Shapiro's Lemma and induction/coinduction functors to relate cohomology over subgroups to the full group.
• Employment of generalized orbits in the group ring and cut-off arguments to control summations and ensure convergence in the $l^p$-norm.
• Use of the duality between $l^p$ and $l^q$ spaces via the Hahn-Banach theorem and adjoint maps to establish the duality isomorphism.
• Adaptation of the proof strategy from the central element case to more general settings like nilpotent and polynomial-growth groups using central series and conjugacy class finiteness.

Experimental results

Research questions

• RQ1 Does reduced $l^p$-homology for a group of type $FP_n$ dualize to reduced $l^q$-cohomology when $\frac{1}{p} + \frac{1}{q} = 1$?
• RQ2 Under what structural conditions on a group of type $FP_n$ does the reduced $l^p$-cohomology vanish for $p > 1$?
• RQ3 Can the vanishing result for groups with a central element of infinite order be extended to groups with infinitely many finite conjugacy classes or nilpotent structure?
• RQ4 How does the presence of a central series with infinite and finite parts affect the $l^p$-cohomology of a nilpotent group?
• RQ5 To what extent do polynomial growth and finite-index nilpotent subgroups imply vanishing of reduced $l^p$-cohomology in $FP_n$ groups?

Key findings

• Reduced $l^p$-homology and $l^q$-cohomology for a group of type $FP_n$ are dual when $\frac{1}{p} + \frac{1}{q} = 1$, via a natural isomorphism induced by the $l^p$-$l^q$ duality.
• For a group of type $FP_n$ with a central element of infinite order, the reduced $l^p$-cohomology vanishes for all $i \leq n$.
• The vanishing result extends to groups with infinitely many finite conjugacy classes, as shown via generalized orbits and cut-off arguments in the group ring.
• Finitely generated infinite nilpotent groups of type $FP_\infty$ have trivial reduced $l^p$-cohomology in all degrees $i \in \mathbb{N}$.
• Groups of polynomial growth, which contain a nilpotent subgroup of finite index, also have trivial reduced $l^p$-cohomology for all $i \in \mathbb{N}$.
• The proof technique relies on constructing controlled summations over orbits of central group ring elements and using norm closure to ensure convergence in $l^p$-spaces.