Local Criteria for Quasirandomness and the Ultraproducts of Quasirandom Groups

نویسنده

  • Yilong Yang
چکیده

A group is called D-quasirandom if all of its nontrivial representations over the complex numbers have dimensions at least D. In this paper, we investigate the extent to which quasirandomness is a local property; more precisely, we study the question of whether a non-principal ultraproduct of a given sequence of quasirandom groups remains quasirandom, and whether the ultraproduct of a sequence of increasingly quasirandom groups becomes minimally almost periodic (i.e., the ultraproduct has no nontrivial finite-dimensional representation at all). We answer this question in the affirmative when the groups in question are simple, quasisimple, semisimple, or when the groups in question have bounded number of conjugacy classes in their cosocles (the intersection of all maximal normal subgroups), or when the groups are arbitrary products of the groups just listed. We also obtain some results in the case of semi-direct products and short exact sequences of quasirandom groups. Our main tools are some variations of the covering number for groups, different kinds of length functions on groups, and the classification of finite simple groups.

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

The poset of hypergraph quasirandomness

Chung and Graham began the systematic study of hypergraph quasirandom properties soon after the foundational results of Thomason and Chung-Graham-Wilson on quasirandom graphs. One feature that became apparent in the early work on hypergraph quasirandomness is that properties that are equivalent for graphs are not equivalent for hypergraphs, and thus hypergraphs enjoy a variety of inequivalent q...

متن کامل

Survey of Quasirandomness in Number Theory

In [9], the author introduced quasirandom permutations, permutations of Zn which map intervals to sets with low discrepancy. Here we show that several natural number-theoretic permutations are quasirandom, some very strongly so. Quasirandomness is established via discrete Fourier analysis and the ErdősTurán inequality, as well as by other means. We apply our results on Sós permutations to make ...

متن کامل

Counting Cycles to Efficiently Certify Sparse Hypergraph Quasirandomness

One surprising property of Chung, Graham, and Wilson’s characterization of dense quasirandom graphs is a polynomial-time verifiable property Cycle4, which states that the number of copies of the cycle of length four is what one would expect in a random graph of the same density. Targeting problems like random k-SAT, this algorithm has been extended in several ways to sparse quasirandomness by s...

متن کامل

Quasirandom Arithmetic Permutations

In [9], the author introduced quasirandom permutations, permutations of Zn which map intervals to sets with low discrepancy. Here we show that several natural number-theoretic permutations are quasirandom, some very strongly so. Quasirandomness is established via discrete Fourier analysis and the ErdősTurán inequality, as well as by other means. We apply our results on Sós permutations to make ...

متن کامل

Hamilton cycles in quasirandom hypergraphs

We show that, for a natural notion of quasirandomness in k-uniform hypergraphs, any quasirandom k-uniform hypergraph on n vertices with constant edge density and minimum vertex degree Ω(nk−1) contains a loose Hamilton cycle. We also give a construction to show that a k-uniform hypergraph satisfying these conditions need not contain a Hamilton `-cycle if k − ` divides k. The remaining values of ...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 2014