The first \(L^{p}\)-cohomology of some finitely generated groups and \(p\)-harmonic functions (Q2499242)

Let \(G\) be a finitely generated infinite group and \(p>1\). For a right \(G\)-module \(M\), \(C^n(G,M)\) denotes the set of functions from \(G^n\) (the Cartesian product of \(G\) \(n\)-times) to \(M\). A chain complex is defined from the sequence, \(C^n(G,M)\), and hence the \(n\)th cohomology of \(G\) with coefficients in \(M\), denoted \(H^n(G,M)\). Of particular interest here is the case where \(M=L^p(G)\), the \(p\)-summable functions on \(G\). A connection is made between the first \(L^p\)-cohomology and \(p\)-harmonic functions on \(G\). The author proves that if \(G\) has polynomial growth, \(d\), the reduced first \(L^p\)-cohomology group is zero. Each non-zero class in \(H^1(G,L^p(G))\) can be represented by an element of \(L^{pd/(d-p)}(G)\) if the growth, \(d\), satisfies: \(d>p\geq 2\). In the last section of the paper the author proves that if \(G\) is a finitely generated nonamenable group and \(1<p\leq q\) then the first \(L^p\)-cohomology of \(G\) is contained in the first \(L^q\)-cohomology. An example is also given to show that this inclusion is not true in case \(G\) is amenable.