Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Relative Ends, l^2 Invariants and Property (T)   We establish a splitting theorem for one-ended groups H<G such that
\tilde{e}(G;H)> 2 and the almost malnormal closure of H is a proper subgroup of
G. This yields splitting theorems for groups G with non-trivial first l^2 Betti
number (\beta^2_1(G)). We verify the Kropholler Conjecture for pairs H < G
satisfying \beta^2_1(G) > \beta^2_1(H). We also prove that every n-dimensional
Poincare duality (PD^n) group containing a PD^(n-1) group H with property (T)
splits over a subgroup commensurable with H.

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The categories of this paper are: physics
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The categories of this paper are: hep-th
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The categories of this paper are: cs
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The categories of this paper are: cond-mat
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The categories of this paper are: acc-phys
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The categories of this paper are: nucl-ex
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The categories of this paper are: quant-ph
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The categories of this paper are: nlin
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The categories of this paper are: hep-ex
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The categories of this paper are: adap-org
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The categories of this paper are: math-ph
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The categories of this paper are: econ
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The categories of this paper are: ao-sci
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The categories of this paper are: hep-lat
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The categories of this paper are: comp-gas
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The categories of this paper are: astro-ph