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Basis-conjugating automorphisms of a free group and associated Lie algebras
Let F_n = denote the free group with generators {x_1,...,x_n}. Nielsen and Magnus described generators for the kernel of the canonical epimorphism from the automorphism group of F_n to the generalExpand
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Group actions and geometric combinatorics in 𝔽 q d $\mathbb{F}_{q}^{d}$
Abstract In this paper we apply a group action approach to the study of Erdős–Falconer-type problems in vector spaces over finite fields and use it to obtain non-trivial exponents for theExpand
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The Fuglede conjecture holds in ℤp × ℤp
In the 70s Bent Fuglede conjectured that if Ω is bounded domain in R, then L(Ω) has an orthogonal basis of exponentials if and only if Ω tiles R by translation. This idea led to much activity andExpand
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Tiling sets and spectral sets over finite fields
We study tiling and spectral sets in vector spaces over prime fields. The classical Fuglede conjecture in locally compact abelian groups says that a set is spectral if and only if it tiles byExpand
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COHOMOLOGY OF UNIFORMLY POWERFUL p-GROUPS
In this paper we will study the cohomology of a family of p-groups associated to Fp-Lie algebras. More precisely we study a category BGrp of p-groups which will be equivalent to the category ofExpand
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Long paths in the distance graph over large subsets of vector spaces over finite fields
Let $E \subset {\Bbb F}_q^d$, the $d$-dimensional vector space over a finite field with $q$ elements. Construct a graph, called the distance graph of $E$, by letting the vertices be the elements ofExpand
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Three-point configurations determined by subsets of $$\mathbb{F}_q^2$$ via the Elekes-Sharir Paradigm
AbstractWe prove that if $$E \subset \mathbb{F}_Q^2$$, q ≡ 3 mod 4, has size greater than $$Cq^{\tfrac{7} {4}}$$, then E determines a positive proportion of all congruence classes of triangles inExpand
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Geometric configurations in the ring of integers modulo $p^{\ell}$
We study variants of the Erd\H os distance problem and dot products problem in the setting of the integers modulo $q$, where $q = p^{\ell}$ is a power of an odd prime.
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Threshold complexes and connections to number theory
In this paper we study quota complexes (or equivalently in the case of scalar weights, threshold complexes) and how the topology of these quota complexes changes as the quota is changed. This problemExpand
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