Harmonic functions of linear growth on solvable groups

Abstract

Abstract. In this work we study the structure of finitely generated groups for which a space of harmonic functions with fixed polynomial growth is finite dimensional. It is conjectured that such groups must be virtually nilpotent (the converse direction to Kleiner's theorem). We prove that this is indeed the case for solvable groups. The investigation is partly motivated by Kleiner's proof for Gromov's theorem on groups of polynomial growth.

Polynomials and harmonic functions on discrete groups

“…In conjunction with the results of [17], this gives the following, which we prove in Subsection 7.3. Corollary 1.9.…”

“…Note that if µ has exponential tail with respect to some finite symmetric generating set then it also has exponential tail with respect to any other finite symmetric generating set, so the notion of having exponential tail does not depend on the specific choice of generating set. The same standing assumptions appeared in [17], where measures with exponential tail were called "smooth". Note that for f of polynomial growth of degree k the assumption that µ has exponential tail implies that In this paper we are mainly interested in the spaces H k (G, µ) of harmonic functions of polynomial growth, defined by…”

“…Proof of Theorem 1.3. The case k = 0 is handled by observing that the constants are the only finite dimensional G-invariant space of bounded harmonic functions; see [17] for details and references. A proof for the case k = 1 appears in [17].…”

“…17 implies that the iterated commutator [ , · · · , ] j induces a multilinear mapα : N/[N, N ] × · · · × N/[N, N ] → N j /N j+1 .It follows from [8, Theorem 10.2.3] that, in the notation of Lemma 4.18, we have N j /N j+1 = α(N/[N, N ]) and H j /(H j ∩ N j+1 ) = α(H/(H ∩ [N, N ])), and so Lemma 4.18 implies that H j /(H j ∩ N j+1 ) has finite index in N j /N j+1 .…”

Conditionally negative type functions on groups acting on regular trees

“…It is fairly easy to check that a group which admits a harmonic function with gradient in C 0 has infinitely many Lipschitz harmonic functions (hence may not be nilpotent, see [CM] or [K,Theorem 1.4]). An argument, which may be found in [MY,§2.2], also shows that the lamplighter (with finite lamp states) on Z has no sublinear harmonic functions (hence no harmonic function with gradient in C 0 ). G. Kozma pointed out to the first named author that this is also the case for solvable Baumslag-Solitar groups.…”