Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userFree flags over local rings and powering of high dimensional expanders
210
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Powering the adjacency matrix of an expander graph results in a better expander of higher degree. In this paper we seek an analogue operation for high-dimensional expanders. We show that the naive approach to powering does not preserve high-dimensional expansion, and define a new power operation, using geodesic walks on quotients of Bruhat-Tits buildings. Applying this operation results in high-dimensional expanders of higher degrees. The crux of the proof is a combinatorial study of flags of free modules over finite local rings. Their geometry describes links in the power complex, and showing that they are excellent expanders implies high dimensional expansion for the power-complex by Garlands local-to-global technique. As an application, we use our power operation to obtain new efficient double samplers.
rate research
Read More
The generalized wreath product of permutation groups is introduced. By means of it we study the schurity problem for S-rings over a cyclic group $G$ and the automorphism groups of them. Criteria for the schurity and non-schurity of the generalized wreath product of two such S-rings are obtained. As a byproduct of the developed theory we prove that $G$ is a Schur group whenever the total number $Omega(n)$ of prime factors of the integer $n=|G|$ is at most 3. Moreover, we describe the structure of a non-schurian S-ring over $G$ when $Omega(n)=4$. The latter result implies in particular that if $n=p^3q$ where $p$ and $q$ are primes, then $G$ is a Schur group.
We present an elementary way to transform an expander graph into a simplicial complex where all high order random walks have a constant spectral gap, i.e., they converge rapidly to the stationary distribution. As an upshot, we obtain new constructions, as well as a natural probabilistic model to sample constant degree high-dimensional expanders. In particular, we show that given an expander graph $G$, adding self loops to $G$ and taking the tensor product of the modified graph with a high-dimensional expander produces a new high-dimensional expander. Our proof of rapid mixing of high order random walks is based on the decomposable Markov chains framework introduced by Jerrum et al.
In this paper, we study the expanding phenomena in the setting of higher dimensional matrix rings. More precisely, we obtain a sum-product estimate for large subsets and show that x+yz, x(y+z) are moderate expanders over the matrix ring, and xy + z + t is strong expander over the matrix rings. These results generalize recent results of Y.D. Karabulut, D. Koh, T. Pham, C-Y. Shen, and the second listed author.
The recently developed theory of Schur rings over a finite cyclic group is generalized to Schur rings over a ring R being a product of Galois rings of coprime characteristics. It is proved that if the characteristic of R is odd, then as in the cyclic group case any pure Schur ring over R is the tensor product of a pure cyclotomic ring and Schur rings of rank 2 over non-fields. Moreover, it is shown that in contrast to the cyclic group case there are non-pure Schur rings over R that are not generalized wreath products.
A commutative ring R has finite rank r, if each ideal of R is generated at most by r elements. A commutative ring R has the r-generator property, if each finitely generated ideal of R can be generated by r elements. Such rings are closely related to Prufer domains. In the present paper we investigate some analogs of these concepts for modules over group rings.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
