arXiv subjects

Gabor Kun

Publications and source records attributed to Gabor Kun.

13 recordsLinked to original sources

arXiv2026

Nonsofic wreath products of residually finite groups

This work builds on the breakthrough of OpenAI in finding the first nonsofic group. We analyze the underlying proof mechanism and find further applications. Let $\Gamma<G$ be such that $\{g\in G:g\Gamma g^{-1}\leq\Gamma\}$ generates $G$ as a group, and suppose that both $\Gamma$ and $G$ have property $(T)$. If $\Gamma$ is not normal, then the generalized wreath product $\bigl(\bigoplus_{G/\Gamma}\mathbb Z/2\mathbb Z\bigr)\rtimes G$ and the group double $G \ast_{\Gamma} G$ are nonsofic. These hypotheses hold for explicit pairs of elementary groups over polynomial and Laurent polynomial rings, in which both groups are residually finite and Kazhdan.

math.GR↗

arXiv2022

Perfect matchings in hyperfinite graphings

We characterize hyperfinite bipartite graphings that admit measurable perfect matchings. In particular, we prove that every regular hyperfinite bipartite graphing admits a measurable perfect matching if it is one-ended or the degree is odd. We give several applications of this result, answering various open questions in the field. For instance, we extend the Lyons--Nazarov theorem by characterizing bipartite Cayley graphs which admit a factor of iid perfect matching, answering the bipartite case of a well-known question of Lyons and Nazarov, popularized by Kechris and Marks. Moreover, we show how our results apply to measurable equidecompositions and, in particular, generalize the recent result of Grabowski, Máthé and Pikhurko on the measurable circle squaring. Our approach applies more generally to rounding measurable perfect fractional matchings.

math.CO↗

arXiv2021

Expanders have a spanning Lipschitz subgraph with large girth

We show that every regular graph with good local expansion has a spanning Lipschitz subgraph with large girth and minimum degree. In particular, this gives a finite analogue of the dynamical solution to the von Neumann problem by Gaboriau and Lyons. We give a new proof and strengthen the Gaboriau-Lyons result, that allows us to answer two questions of Monod about geometric random subgroups. Our finite theorems are kind of converse to the theorem of Bourgain and Gamburd showing that large girth implies expansion for Cayley graphs of SL_2(F_p). We apply these to the regular case of Thomassen's conjecture stating that every finite graph with large average degree has a subgraph with large girth and average degree. Our main tool is an infinite version of the Lovasz Local Lemma developed in this paper.

math.GR↗

arXiv2021

On Gardner's conjecture

Gardner conjectured that if two bounded measurable sets $A,B \subset \mathbb{R}^n$ are equidecomposable by a set of isometries $Γ$ generating an amenable group then $A$ and $B$ admit a measurable equidecomposition by all isometries. Cieśla and Sabok asked if there is a measurable equidecomposition using isometries only in the group generated by $Γ$. We answer this question negatively.

math.MG↗

arXiv2020

Expander spanning subgraphs with large girth

We conjecture that finite graphs with positive Cheeger constant admit a spanning subgraph with positive Cheeger constant and girth proportional to the diameter. We prove this conjecture for regular expander graphs with large expansion. Our proof relies on the Local Lemma.

math.CO↗

arXiv2019

On sofic approximations of Property (T) groups

We prove Bowen's conjecture that every sequence of finite graphs that locally converges to the Cayley graph of a countably infinite group with Kazhdan Property (T) is essentially a vertex-disjoint union of expander graphs. We characterize graph sequences that are essentially a vertex-disjoint union of expander graphs in terms of the Markov operator.

math.CO↗

arXiv2019

Inapproximability of actions and Kazhdan's property (T)

Let G be a countable group with Kazhdan property (T) and let G be a finitely generated group. We prove that if a sofic approximation of G x D has the property that its spanning subgraphs consisting of the G-labeled edges form an expander sequence, then D is locally embeddable into finite groups (LEF). Consequently, if D is not LEF, then every almost free p.m.p. action of G x D whose restriction to G is ergodic fails to weakly contain any sequence of finite labeled graphs; in particular, such an action is not a local-global limit of finite graphs. We also prove that, when D is amenable, the existence of an expander sofic approximation of G x D forces D to be LEF. The main technical input is a self-improvement theorem for almost automorphisms of expander sofic approximations of property (T) groups. It implies that the Hamming-distance clusters of sufficiently accurate almost automorphisms carry a natural finite group structure.

math.GR↗

arXiv2013

Constraints, MMSNP and expander relational structures

We give a poly-time construction for a combinatorial classic known as Sparse Incomparability Lemma, studied by Erdos, Lovasz, Nesetril, Rodl and others: We show that every Constraint Satisfaction Problem is poly-time equivalent to its restriction to structures with large girth. This implies that the complexity classes CSP and Monotone Monadic Strict NP introduced by Feder and Vardi are computationally equivalent. The technical novelty of the paper is a concept of expander relations and a new type of product for relational structures: a generalization of the zig-zag product, the twisted product.

math.CO↗

arXiv2012

Lattice Sparsification and the Approximate Closest Vector Problem

We give a deterministic algorithm for solving the (1+eps)-approximate Closest Vector Problem (CVP) on any n dimensional lattice and any norm in 2^{O(n)}(1+1/eps)^n time and 2^n poly(n) space. Our algorithm builds on the lattice point enumeration techniques of Micciancio and Voulgaris (STOC 2010) and Dadush, Peikert and Vempala (FOCS 2011), and gives an elegant, deterministic alternative to the "AKS Sieve" based algorithms for (1+eps)-CVP (Ajtai, Kumar, and Sivakumar; STOC 2001 and CCC 2002). Furthermore, assuming the existence of a poly(n)-space and 2^{O(n)} time algorithm for exact CVP in the l_2 norm, the space complexity of our algorithm can be reduced to polynomial. Our main technical contribution is a method for "sparsifying" any input lattice while approximately maintaining its metric structure. To this end, we employ the idea of random sublattice restrictions, which was first employed by Khot (FOCS 2003) for the purpose of proving hardness for Shortest Vector Problem (SVP) under l_p norms.

cs.DS↗

arXiv2008

Cops and robbers in random graphs

We consider the pursuit and evasion game on finite, connected, undirected graphs known as cops and robbers. Meyniel conjectured that for every graph on n vertices a rootish number of cops can win the game. We prove that this holds up to a log(n) factor for random graphs G(n,p) if p is not very small, and this is close to be tight unless the graph is very dense. We analyze the area-defending strategy (used by Aigner in case of planar graphs) and show examples where it can not be too efficient.

math.CO↗

arXiv2007

NP by means of lifts and shadows

We show that every NP problem is polynomially equivalent to a simple combinatorial problem: the membership problem for a special class of digraphs. These classes are defined by means of shadows (projections) and by finitely many forbidden colored (lifted) subgraphs. Our characterization is motivated by the analysis of syntactical subclasses with the full computational power of NP, which were first studied by Feder and Vardi. Our approach applies to many combinatorial problems and it induces the characterization of coloring problems (CSP) defined by means of shadows. This turns out to be related to homomorphism dualities. We prove that a class of digraphs (relational structures) defined by finitely many forbidden colored subgraphs (i.e. lifted substructures) is a CSP class if and only if all the the forbidden structures are homomorphically equivalent to trees. We show a surprising richness of coloring problems when restricted to most frequent graph classes. Using results of Nešetřil and Ossona de Mendez for bounded expansion classes (which include bounded degree and proper minor closed classes) we prove that the restriction of every class defined as the shadow of finitely many colored subgraphs equals to the restriction of a coloring (CSP) class.

cs.CC↗

arXiv2007

Forbidden lists (NP and CSP for combinatorialists)

We present a definition of the class NP in combinatorial context as the set of languages of structures defined by finitely many forbidden lifted substructures. We apply this to special syntactically defined subclasses and show how they correspond to naturally defined (and intensively studied) combinatorial problems. We show that some types of combinatorial problems like edge colorings and graph decompositions express the full computational power of the class NP. We then characterize Constraint Satisfaction Problems (i.e. H-coloring problems) which are expressible by finitely many forbidden lifted substructures. This greatly simplifies and generalizes the earlier attempts to characterize this problem. As a corollary of this approach we perhaps find a proper setting of Feder and Vardi analysis of CSP languages within the class MMSNP.

math.CO↗

arXiv2005

Prime values of reducible polynomials, II

The Schinzel hypothesis claims (but it seems hopeless to prove) that any irreducible Q[x] polynomial without a constant factor assumes infinitely many prime values at integer places. On the other hand, it is easy to see that a reducible Q[x] polynomial can have only finitely many such places. In this paper we prove that a reducible Z[x] polynomial of degree n (where n is not 4 or 5) can have at most n+2 such places, and there exist examples with n+1 places. If the Schinzel hypothesis and the k-prime-tuple conjecture are true, then there are also polynomials with n+2 such places. (For n=4 or 5 the maximum possible value is 8.) If a Z[x] polynomial is the product of two non-constant integer-valued Q[x] polynomials, then there are at most 1.87234...n+o(n) such places. Even in this case, the number of integer places with positive prime values is at most n. More generally, if f=gh \in R[x] is a product of two non-constant real polynomials, then the number of real places x such that |g(x)|=1 or |h(x)=1|, while f(x)>1, is at most n. For a natural complex version of this last statement we give a counterexample. Furthermore, we briefly consider the generalized problem of assuming values in a fixed finite set of integers.

math.NT↗