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Register a new userCoboundary and cosystolic expansion from strong symmetry
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Coboundary and cosystolic expansion are notions of expansion that generalize the Cheeger constant or edge expansion of a graph to higher dimensions. The classical Cheeger inequality implies that for graphs edge expansion is equivalent to spectral expansion. In higher dimensions this is not the case: a simplicial complex can be spectrally expanding but not have high dimensional edge-expansion. The phenomenon of high dimensional edge expansion in higher dimensions is much more involved than spectral expansion, and is far from being understood. In particular, prior to this work, the only known bounded degree cosystolic expanders known were derived from the theory of buildings that is far from being elementary. In this work we study high dimensional complexes which are {em strongly symmetric}. Namely, there is a group that acts transitively on top dimensional cells of the simplicial complex [e.g., for graphs it corresponds to a group that acts transitively on the edges]. Using the strong symmetry, we develop a new machinery to prove coboundary and cosystolic expansion.
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Following Gromov, the coboundary expansion of building-like complexes is studied. In particular, it is shown that for any $n geq 1$, there exists a constant $epsilon(n)>0$ such that for any $0 leq k <n$ the $k$-th coboundary expansion constant of any $n$-dimensional spherical building is at least $epsilon(n)$.
We survey some principal results and open problems related to colorings of algebraic and geometric objects endowed with symmetries.
In this work we present a new local to global criterion for proving a form of high dimensional expansion, which we term cosystolic expansion. Applying this criterion on Ramanujan complexes, yields for every dimension, an infinite family of bounded degree complexes with the topological overlapping property. This answer affirmatively an open question raised by Gromov.
We study the adsorption of Li to graphene flakes described as aromatic molecules. Surprisingly the out of plane deformation is much stronger for the double adsorption from both sides to the same ring than for a single adsorption, although a symmetric solution seems possible. We thus have an interesting case of spontaneous symmetry breaking. While we cannot rule out a Jahn Teller deformation with certainty, this explanation seems unlikely and other options are discussed. We find a similar behavior for Boron-Nitrogen sheets, and also for other light alkalines.
We build, from the collection of all groups of unitriangular matrices, Hopf monoids in Joyals category of species. Such structure is carried by the collection of class function spaces on those groups, and also by the collection of superclass function spaces, in the sense of Diaconis and Isaacs. Superclasses of unitriangular matrices admit a simple description from which we deduce a combinatorial model for the Hopf monoid of superclass functions, in terms of the Hadamard product of the Hopf monoids of linear orders and of set partitions. This implies a recent result relating the Hopf algebra of superclass functions on unitriangular matrices to symmetric functions in noncommuting variables. We determine the algebraic structure of the Hopf monoid: it is a free monoid in species, with the canonical Hopf structure. As an application, we derive certain estimates on the number of conjugacy classes of unitriangular matrices.
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