Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The first uniformly finite homology group with coefficients in
  $\mathbb{Z}$ and a characterisation of its vanishing in the transitive case   We study the first uniformly finite homology group of Block and Weinberger
for uniformly locally finite graphs, with coefficients in $\mathbb{Z}$ and
$\mathbb{Z}_2$. When the graph is a tree, or coefficients are in
$\mathbb{Z}_2$, a characterisation of the group is obtained. In the general
case, we describe three phenomena that entail non-vanishing of the group; their
disjunction is shown to also be necessary for non-vanishing in the case of
transitive graphs.

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The categories of this paper are: gr-qc
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The categories of this paper are: q-alg
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The categories of this paper are: nucl-ex
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The categories of this paper are: cmp-lg
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The categories of this paper are: q-fin
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The categories of this paper are: adap-org
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The categories of this paper are: chao-dyn
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The categories of this paper are: bayes-an
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The categories of this paper are: chem-ph
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The categories of this paper are: q-bio
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The categories of this paper are: nucl-th
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The categories of this paper are: astro-ph
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The categories of this paper are: funct-an
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The categories of this paper are: stat
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The categories of this paper are: alg-geom
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The categories of this paper are: hep-ph