An official website of the United States government
We first give simplified and corrected accounts of some results in work by Pillay (2017) on compactifications of pseudofinite groups. For instance, we use a classical theorem of Turing (1938) to give a simplified proof that any definable compactification of a pseudofinite group has an abelian connected component. We then discuss the relationship between Turing’s work, the Jordan–Schur theorem, and a (relatively) more recent result of Kazhdan (1982) on approximate homomorphisms, and we use this to widen our scope from finite groups to amenable groups. In particular, we develop a suitable continuous logic framework for dealing with definable homomorphisms from pseudoamenable groups to compact Lie groups. Together with the stabilizer theorems of Hrushovski (2012) and Montenegro et al. (2020), we obtain a uniform (but non-quantitative) analogue of Bogolyubov’s lemma for sets of positive measure in discrete amenable groups. We conclude with brief remarks on the case of amenable topological groups.
more »
« less
List-Decoding Homomorphism Codes with Arbitrary Codomains
The codewords of the homomorphism code aHom(G,H) are the affine homomorphisms between two finite groups, G and H, generalizing Hadamard codes. Following the work of Goldreich-Levin (1989), Grigorescu et al. (2006), Dinur et al. (2008), and Guo and Sudan (2014), we further expand the range of groups for which local list-decoding is possible up to mindist, the minimum distance of the code. In particular, for the first time, we do not require either G or H to be solvable. Specifically, we demonstrate a poly(1/epsilon) bound on the list size, i. e., on the number of codewords within distance (mindist-epsilon) from any received word, when G is either abelian or an alternating group, and H is an arbitrary (finite or infinite) group. We conjecture that a similar bound holds for all finite simple groups as domains; the alternating groups serve as the first test case. The abelian vs. arbitrary result permits us to adapt previous techniques to obtain efficient local list-decoding for this case. We also obtain efficient local list-decoding for the permutation representations of alternating groups (the codomain is a symmetric group) under the restriction that the domain G=A_n is paired with codomain H=S_m satisfying m < 2^{n-1}/sqrt{n}. The limitations on the codomain in the latter case arise from severe technical difficulties stemming from the need to solve the homomorphism extension (HomExt) problem in certain cases; these are addressed in a separate paper (Wuu 2018). We introduce an intermediate "semi-algorithmic" model we call Certificate List-Decoding that bypasses the HomExt bottleneck and works in the alternating vs. arbitrary setting. A certificate list-decoder produces partial homomorphisms that uniquely extend to the homomorphisms in the list. A homomorphism extender applied to a list of certificates yields the desired list.
more »
« less
- Award ID(s):
- 1718902
- PAR ID:
- 10179666
- Date Published:
- Journal Name:
- Leibniz international proceedings in informatics
- Volume:
- 116
- ISSN:
- 1868-8969
- Page Range / eLocation ID:
- 29:1-29:18
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Abstract. Using totally symmetric sets, Chudnovsky–Kordek–Li–Partin gave a superexponential lower bound on the cardinality of non-abelian finite quotients of the braid group. In this paper, we develop new techniques using multiple totally symmetric sets to count elements in non-abelian finite quotients of the braid group. Using these techniques, we improve the lower bound found by Chudnovsky et al. We exhibit totally symmetric sets in the virtual and welded braid groups and use our new techniques to find superexponential bounds for the finite quotients of the virtual and welded braid groups.more » « less
-
We consider the task of locally correcting, and locally list-correcting, multivariate linear functions over the domain {0,1}n over arbitrary fields and more generally Abelian groups. Such functions form error-correcting codes of relative distance 1/2 and we give local-correction algorithms correcting up to nearly 1/4-fraction errors making O(logn) queries. This query complexity is optimal up to poly(loglogn) factors. We also give local list-correcting algorithms correcting (1/2 − ε)-fraction errors with Oε(logn) queries. These results may be viewed as natural generalizations of the classical work of Goldreich and Levin whose work addresses the special case where the underlying group is ℤ2. By extending to the case where the underlying group is, say, the reals, we give the first non-trivial locally correctable codes (LCCs) over the reals (with query complexity being sublinear in the dimension (also known as message length)). Previous works in the area mostly focused on the case where the domain is a vector space or a group and this lends to tools that exploit symmetry. Since our domains lack such symmetries, we encounter new challenges whose resolution may be of independent interest. The central challenge in constructing the local corrector is constructing “nearly balanced vectors” over {−1,1}n that span 1n — we show how to construct O(logn) vectors that do so, with entries in each vector summing to ±1. The challenge to the local-list-correction algorithms, given the local corrector, is principally combinatorial, i.e., in proving that the number of linear functions within any Hamming ball of radius (1/2−ε) is Oε(1). Getting this general result covering every Abelian group requires integrating a variety of known methods with some new combinatorial ingredients analyzing the structural properties of codewords that lie within small Hamming balls.more » « less
-
We consider the task of locally correcting, and locally list-correcting, multivariate linear functions over the domain {0,1}n over arbitrary fields and more generally Abelian groups. Such functions form error-correcting codes of relative distance 1/2 and we give local-correction algorithms correcting up to nearly 1/4-fraction errors making O(logn) queries. This query complexity is optimal up to poly(loglogn) factors. We also give local list-correcting algorithms correcting (1/2 − ε)-fraction errors with Oε(logn) queries. These results may be viewed as natural generalizations of the classical work of Goldreich and Levin whose work addresses the special case where the underlying group is ℤ2. By extending to the case where the underlying group is, say, the reals, we give the first non-trivial locally correctable codes (LCCs) over the reals (with query complexity being sublinear in the dimension (also known as message length)). Previous works in the area mostly focused on the case where the domain is a vector space or a group and this lends to tools that exploit symmetry. Since our domains lack such symmetries, we encounter new challenges whose resolution may be of independent interest. The central challenge in constructing the local corrector is constructing “nearly balanced vectors” over {−1,1}n that span 1n — we show how to construct O(logn) vectors that do so, with entries in each vector summing to ±1. The challenge to the local-list-correction algorithms, given the local corrector, is principally combinatorial, i.e., in proving that the number of linear functions within any Hamming ball of radius (1/2−ε) is Oε(1). Getting this general result covering every Abelian group requires integrating a variety of known methods with some new combinatorial ingredients analyzing the structural properties of codewords that lie within small Hamming balls.more » « less
-
We show that a -stable set in a finite group can be approximated, up to given error , by left cosets of a subgroup of index . This improves the bound in a similar result of Terry and Wolf on stable arithmetic regularity in finite abelian groups, and leads to a quantitative account of work of the author, Pillay, and Terry on stable sets in arbitrary finite groups. We also prove an analogous result for finite stable sets of small tripling in arbitrary groups, which provides a quantitative version of recent work by Martin-Pizarro, Palacín, and Wolf. Our proofs use results on VC-dimension, and a finitization of model-theoretic techniques from stable group theory.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- The DOI is not currently available.
