Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Characterisations of algebraic properties of groups in terms of harmonic
  functions   We prove various results connecting structural or algebraic properties of
graphs and groups to conditions on their spaces of harmonic functions. In
particular: we show that a group with a finitely supported symmetric measure
has a finite-dimensional space of harmonic functions if and only if it is
virtually cyclic; we present a new proof of a result of V. Trofimov that an
infinite vertex-transitive graph admits a non-constant harmonic function; we
give a new proof of a result of T. Ceccherini-Silberstein, M. Coornaert and J.
Dodziuk that the Laplacian on an infinite, connected, locally finite graph is
surjective; and we show that the positive harmonic functions on a non-virtually
nilpotent linear group span an infinite-dimensional space.

27
The categories of this paper are: comp-gas
21
The categories of this paper are: adap-org
37
The categories of this paper are: supr-con
0
The categories of this paper are: hep-ph
8
The categories of this paper are: hep-ex
36
The categories of this paper are: cmp-lg
34
The categories of this paper are: chem-ph
19
The categories of this paper are: eess
24
The categories of this paper are: dg-ga
35
The categories of this paper are: mtrl-th
29
The categories of this paper are: funct-an
5
The categories of this paper are: gr-qc
4
The categories of this paper are: cond-mat
14
The categories of this paper are: math-ph
7
The categories of this paper are: hep-th
3
The categories of this paper are: physics