Yuval Ginosar

The majority action on infinite graphs: strings and puppets

Abstract We consider the following dynamic process on the 0–1 colourings of the vertices of a graph. The initial state is an arbitrary colouring, and the state at time t +1 is determined by assigning to each vertex the colour of the majority of its neighbours at time t (in case of a tie, the vertex retains its own colour at time t). It is known that if the graph is finite then the process either reaches a fixed colouring or becomes periodic with period two. Here we show that an infinite (locally finite) graph displays the same behaviour locally, provided that the graph satisfies a certain condition which, roughly speaking, imposes an upper bound on the growth rate of the graph. Among the graphs obeying this condition are some that are most common in applications, such as the grid graph in two or more dimensions. We also extend the analysis to more general dynamic processes, and compare our results to the seminal work of Moran in this area.

Isotropy in group cohomology

The analogue of Lagrangians for symplectic forms over finite groups is studied, motivated by the fact that symplectic G-forms with a normal Lagrangian N<G are in one-to-one correspondence, up to inflation, with bijective 1-cocycle data on the quotients G/N. This yields a method to construct groups of central type from such quotients, known as Involutive Yang-Baxter groups. Another motivation for the search of normal Lagrangians comes from a non-commutative generalization of Heisenberg liftings which require normality. Although it is true that symplectic forms over finite nilpotent groups always admit Lagrangians, we exhibit an example where none of these subgroups is normal. However, we prove that symplectic forms over nilpotent groups always admit normal Lagrangians if all their p-Sylow subgroups are of order less than p^8.

On a conjecture of Moore

This can be regarded as a generalization of Serre’s Theorem that every torsion-free group of finite virtual cohomological dimension has finite cohomological dimension (see [2]). For suppose that G is torsion-free and that H has cohomological dimension n < co. Then the nth syzygy in any projective resolution of Z over ZG is projective as ZH-module, and Moore’s Conjecture implies at once that it is also projective as ZG-module. One of the special cases of the conjecture which we prove here also implies Serre’s Theorem. It is natural to generalize the conjecture, because, in addition to group rings, it makes perfect sense for crossed products and in fact for strongly group-graded rings. Recall that a G-graded ring is a ring A with a direct sum decomposition

Hereditary and maximal crossed product orders

We first deal with classical crossed products S f * G, where G is a finite group acting on a Dedekind domain S and S G (the G-invariant elements in S) a DVR, admitting a separable residue fields extension. Here f : G x G → S* is a 2-cocycle. We prove that S f * G is hereditary if and only if S/Jac(S) f * G is semi-simple. In particular, the heredity property may hold even when S/S G is not tamely ramified (contradicting standard textbook references). For an arbitrary Krull domain S, we use the above to prove that under the same separability assumption, S f * G is a maximal order if and only if its height one prime ideals are extended from S. We generalize these results by dropping the residual separability assumptions. An application to non-commutative unique factorization rings is also presented.

A graph-theoretic approach for comparing dimensions of components in simply-graded algebras

Any simple group-grading of a finite dimensional complex algebra induces a natural family of digraphs. We prove that

A SEPARABLE DEFORMATION OF THE QUATERNION GROUP ALGEBRA

|E\circ E^{\text{op}}\cup E^{\text{op}}\circ E|\geq |E|

MAXIMAL CROSSED PRODUCT ORDERS OVER DISCRETE VALUATION RINGS

for any digraph

Induction from Elementary Abelian Subgroups

\Gamma =(V,E)

Estrada index and Chebyshev polynomials

without parallel edges, and deduce that for any simple group-grading, the dimension of the trivial component is maximal.

On groups of I-type and involutive Yang–Baxter groups

The Donald-Flanigan conjecture asserts that for any finite group G and any field k, the group algebra kG can be deformed to a separable algebra. The minimal unsolved instance, namely the quaternion group Q8 over a field k of characteristic 2 was considered as a counterexample. We present here a separable deformation of kQ8. In a sense, the conjecture for any finite group is open again.