Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Ramanujan Property for Simplicial Complexes   Let $G$ be a topological group acting on a simplicial complex $\mathcal{X}$
satisfying some mild assumptions. For example, consider a $k$-regular tree and
its automorphism group, or more generally, a regular affine Bruhat-Tits
building and its automorphism group. We define and study various types of
high-dimensional spectra of quotients of $\mathcal{X}$ by subgroups of $G$.
These spectra include the spectrum of many natural operators associated with
the quotients, e.g. the high-dimensional Laplacians.
  We prove a theorem in the spirit of the Alon-Boppana Theorem, leading to a
notion of Ramanujan quotients of $\mathcal{X}$. Ramanujan $k$-regular graphs
and Ramanuajn complexes in the sense of Lubotzky, Samuels and Vishne are
Ramanujan in dimension $0$ according to our definition (for $\mathcal{X}$, $G$
suitably chosen). We give a criterion for a quotient of $\mathcal{X}$ to be
Ramanujan which is phrased in terms of representations of $G$, and use it,
together with deep results about automorphic representations, to show that
affine buildings of inner forms of $\mathbf{GL}_n$ over local fields of
positive characteristic admit infinitely many quotients which are Ramanujan in
all dimensions. The Ramanujan (in dimension $0$) complexes constructed by
Lubotzky, Samuels and Vishne arise as a special case of our construction. Our
construction also gives rise to Ramanujan graphs which are apparently new.
  Other applications are also discussed. For example, we show that there are
non-isomorphic simiplicial complexes which are isospectral in all dimensions.

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The categories of this paper are: econ
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The categories of this paper are: nucl-th
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The categories of this paper are: solv-int
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The categories of this paper are: dg-ga
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The categories of this paper are: cmp-lg
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The categories of this paper are: acc-phys
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The categories of this paper are: q-alg
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The categories of this paper are: atom-ph
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The categories of this paper are: adap-org
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The categories of this paper are: patt-sol
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The categories of this paper are: stat
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The categories of this paper are: quant-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: plasm-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: chem-ph