IA Scholar Query: On Convex Geometric Graphs with no k + 1 Pairwise Disjoint Edges.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgWed, 23 Nov 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Knots, minimal surfaces and J-holomorphic curves
https://scholar.archive.org/work/jxpgkptcxzhltcsxxsrvo3lkxm
Let K be a knot in the 3-sphere, viewed as the ideal boundary of hyperbolic 4-space ℍ^4. We prove that the number of minimal discs in ℍ^4 with ideal boundary K is a knot invariant. I.e. the number is finite and doesn't change under isotopies of K. In fact this gives a family of knot invariants, indexed by an integer describing the extrinsic topology of how the disc sits in ℍ^4. These invariants can be seen as Gromov–Witten invariants counting J-holomorphic discs in the twistor space Z of ℍ^4. Whilst Gromov–Witten theory suggests the general scheme for defining the invariants, there are substantial differences in how this must be carried out in our situation. These are due to the fact that the geometry of both ℍ^4 and Z becomes singular at infinity, and so the J-holomorphic curve equation is degenerate, rather than elliptic, at the boundary. This means that both the Fredholm and compactness arguments involve completely new features.Joel Finework_jxpgkptcxzhltcsxxsrvo3lkxmWed, 23 Nov 2022 00:00:00 GMTAn introduction to diagram groups
https://scholar.archive.org/work/lqayci4ftjdi5hkml3vbyvleqa
To every semigroup presentation 𝒫= ⟨Σ|ℛ⟩ and every baseword w ∈Σ^+ can be associated a diagram group D(𝒫,w), defined as the fundamental group of the so-called Squier complex S(𝒫,w). Roughly speaking, D(𝒫,w) encodes the lack of asphericity of 𝒫. Examples of diagram groups include Thompson's group F, the lamplighter group ℤ≀ℤ, the pure planar braid groups, and various right-angled Artin groups. This survey aims at summarising what is known about the family of diagram groups.Anthony Genevoiswork_lqayci4ftjdi5hkml3vbyvleqaTue, 22 Nov 2022 00:00:00 GMTMassively Parallel Computation on Embedded Planar Graphs
https://scholar.archive.org/work/kruzjpzirfbsdlveqiebknildi
Many of the classic graph problems cannot be solved in the Massively Parallel Computation setting (MPC) with strongly sublinear space per machine and o(log n) rounds, unless the 1-vs-2 cycles conjecture is false. This is true even on planar graphs. Such problems include, for example, counting connected components, bipartition, minimum spanning tree problem, (approximate) shortest paths, and (approximate) diameter/radius. In this paper, we show a way to get around this limitation. Specifically, we show that if we have a "nice" (for example, straight-line) embedding of the input graph, all the mentioned problems can be solved with O(n^2/3+ϵ) space per machine in O(1) rounds. In conjunction with existing algorithms for computing the Delaunay triangulation, our results imply an MPC algorithm for exact Euclidean minimum spanning thee (EMST) that uses O(n^2/3 + ϵ) space per machine and finishes in O(1) rounds. This is the first improvement over a straightforward use of the standard Borůvka's algorithm with the Dauleanay triangulation algorithm of Goodrich [SODA 1997] which results in Θ(log n) rounds. This also partially negatively answers a question of Andoni, Nikolov, Onak, and Yaroslavtsev [STOC 2014], asking for lower bounds for exact EMST. We extend our algorithms to work with embeddings consisting of curves that are not "too squiggly" (as formalized by the total absolute curvature). We do this via a new lemma which we believe is of independent interest and could be used to parameterize other geometric problems by the total absolute curvature. We also state several open problems regarding massively parallel computation on planar graphs.Jacob Holm, Jakub Tětekwork_kruzjpzirfbsdlveqiebknildiMon, 21 Nov 2022 00:00:00 GMTCircle homeomorphisms with square summable diamond shears
https://scholar.archive.org/work/3neturwvbff47jn7zdu7ilj2bi
We introduce and study ℓ^2 spaces of homeomorphisms of the circle (up to Möbius transformations) with respect to two modular coordinates, namely shears and diamond shears along the edges of the Farey tessellation. Shears and diamond shears are related combinatorially, and the latter is closely related to the logΛ-lengths of decorated universal Teichmüller space introduced by R. Penner. We compare these new classes to the Weil-Petersson class and Hölder classes of circle homeomorphisms. We also express the Weil-Petersson metric tensor and symplectic form in terms of infinitesimal shears and diamond shears.Dragomir Šarić, Yilin Wang, Catherine Wolframwork_3neturwvbff47jn7zdu7ilj2biMon, 21 Nov 2022 00:00:00 GMTCyclic hyperbolicity in CAT(0) cube complexes
https://scholar.archive.org/work/xekm54igizhc7ornlmjcwo43pu
It is known that a cocompact special group G does not contain ℤ×ℤ if and only if it is hyperbolic; and it does not contain 𝔽_2 ×ℤ if and only if it is toric relatively hyperbolic. Pursuing in this direction, we show that G does not contain 𝔽_2 ×𝔽_2 if and only if it is weakly hyperbolic relative to cyclic subgroups, or cyclically hyperbolic for short. This observation motivates the study of cyclically hyperbolic groups, which we initiate in the class of groups acting geometrically on CAT(0) cube complexes. Given such a group G, we first prove a structure theorem: G virtually splits as the direct sum of a free abelian group and an acylindrically hyperbolic cubulable group. Next, we prove a strong Tits alternative: every subgroup H ≤ G either is virtually abelian or it admits a series H=H_0 H_1 ⋯ H_k where H_k is acylindrically hyperbolic and where H_i/H_i+1 is finite or free abelian. As a consequence, G is SQ-universal and it cannot contain subgroups such that products of free groups and virtually simple groups.Anthony Genevoiswork_xekm54igizhc7ornlmjcwo43puSun, 20 Nov 2022 00:00:00 GMTStructure of conjugacy classes in Coxeter groups
https://scholar.archive.org/work/24vkluqs5nddbcn72d5au2tn3q
This paper gives a definitive solution to the problem of describing conjugacy classes in arbitrary Coxeter groups in terms of cyclic shifts. Let (W,S) be a Coxeter system. A cyclic shift of an element w∈ W is a conjugate of w of the form sws for some simple reflection s∈ S such that ℓ_S(sws)≤ℓ_S(w). The cyclic shift class of w is then the set of elements of W that can be obtained from w by a sequence of cyclic shifts. Given a subset K⊆ S such that W_K:=⟨ K⟩⊆ W is finite, we also call two elements w,w'∈ W K-conjugate if w,w' normalise W_K and w'=w_0(K)ww_0(K), where w_0(K) is the longest element of W_K. Let 𝒪 be a conjugacy class in W, and let 𝒪^min be the set of elements of minimal length in 𝒪. Then 𝒪^min is the disjoint union of finitely many cyclic shift classes C_1,...,C_k. We define the structural conjugation graph associated to 𝒪 to be the graph with vertices C_1,...,C_k, and with an edge between distinct vertices C_i,C_j if they contain representatives u∈ C_i and v∈ C_j such that u,v are K-conjugate for some K⊆ S. In this paper, we compute explicitely the structural conjugation graph associated to any conjugacy class in W, and show in particular that it is connected (that is, any two conjugate elements of W differ only by a sequence of cyclic shifts and K-conjugations). Along the way, we obtain several results of independent interest, such as a description of the centraliser of an infinite order element w∈ W, as well as the existence of natural decompositions of w as a product of a "torsion part" and of a "straight part", with useful properties.Timothée Marquiswork_24vkluqs5nddbcn72d5au2tn3qFri, 18 Nov 2022 00:00:00 GMTMean-field limit of non-exchangeable systems
https://scholar.archive.org/work/spdwnfdkpredpknwjg2mlih2sq
This paper deals with the derivation of the mean-field limit for multi-agent systems on a large class of sparse graphs. More specifically, the case of non-exchangeable multi-agent systems consisting of non-identical agents is addressed. The analysis does not only involve PDEs and stochastic analysis but also graph theory through a new concept of limits of sparse graphs (extended graphons) that reflect the structure of the connectivities in the network and has critical effects on the collective dynamics. In this article some of the main restrictive hypothesis in the previous literature on the connectivities between the agents (dense graphs) and the cooperation between them (symmetric interactions) are removed.Pierre-Emmanuel Jabin, David Poyato, Juan Solerwork_spdwnfdkpredpknwjg2mlih2sqFri, 18 Nov 2022 00:00:00 GMTCrossing and intersecting families of geometric graphs on point sets
https://scholar.archive.org/work/a34faos5gveepgvillugrxygfe
Let S be a set of n points in the plane in general position. Two line segments connecting pairs of points of S cross if they have an interior point in common. Two vertex disjoint geometric graphs with vertices in S cross if there are two edges, one from each graph, which cross. A set of vertex disjoint geometric graphs with vertices in S is called mutually crossing if any two of them cross. We show that there exists a constant c such that from any family of n mutually crossing triangles, one can always obtain a family of at least n^c mutually crossing 2-paths (each of which is the result of deleting an edge from one of the triangles) and then provide an example that implies that c cannot be taken to be larger than 2/3. For every n we determine the maximum number of crossings that a Hamiltonian cycle on a set of n points might have. Next, we construct a point set whose longest perfect matching contains no crossings. We also consider edges consisting of a horizontal and a vertical line segment joining pairs of points of S, which we call elbows, and prove that in any point set S there exists a family of ⌊ n/4 ⌋ vertex disjoint mutually crossing elbows. Additionally, we show a point set that admits no more than n/3 mutually crossing elbows. Finally we study intersecting families of graphs, which are not necessarily vertex disjoint. A set of edge disjoint graphs with vertices in S is called an intersecting family if for any two graphs in the set we can choose an edge in each of them such that they cross. We prove a conjecture by Lara and Rubio-Montiel, namely, that any set S of n points in general position admits a family of intersecting triangles with a quadratic number of elements. Some other results are obtained throughout this work.José Luis Álvarez-Rebollarwork_a34faos5gveepgvillugrxygfeThu, 17 Nov 2022 00:00:00 GMTInfinite-dimensional uniform polyhedra
https://scholar.archive.org/work/wh2pzopopfa3rmhke5iltlp7yi
Uniform covers with a finite-dimensional nerve are rare (i.e., do not form a cofinal family) in many separable metric spaces of interest. To get hold on uniform homotopy properties of these spaces, a reasonably behaved notion of an infinite-dimensional metric polyhedron is needed; a specific list of desired properties was sketched by J. R. Isbell in a series of publications in 1959-64. In this paper we construct what appears to be the desired theory of uniform polyhedra; incidentally, considerable information about their metric and Lipschitz properties is obtained.Sergey A. Melikhovwork_wh2pzopopfa3rmhke5iltlp7yiThu, 17 Nov 2022 00:00:00 GMTLocally normal subgroups and ends of locally compact Kac-Moody groups
https://scholar.archive.org/work/i2cewamf3zdexns3wzqqh57bre
A locally normal subgroup in a topological group is a subgroup whose normaliser is open. In this paper, we provide a detailed description of the large-scale structure of closed locally normal subgroups of complete Kac-Moody groups over finite fields. Combining that description with the main result from arXiv:2111.07066, we show that under mild assumptions, if the Kac-Moody group is one-ended (a property that is easily determined from the generalised Cartan matrix), then it is locally indecomposable, which means that no open subgroup decomposes as a nontrivial direct product.Pierre-Emmanuel Caprace, Timothée Marquis, Colin D. Reidwork_i2cewamf3zdexns3wzqqh57breThu, 17 Nov 2022 00:00:00 GMTCovering and packing with homothets of limited capacity
https://scholar.archive.org/work/uc5owpu35nbdjhcfxstve5zxti
This work revolves around the two following questions: Given a convex body C⊂ℝ^d, a positive integer k and a finite set S⊂ℝ^d (or a finite μ Borel measure in ℝ^d), how many homothets of C are required to cover S if no homothet is allowed to cover more than k points of S (or have measure more than k)? how many homothets of C can be packed if each of them must cover at least k points of S (or have measure at least k)? We prove that, so long as S is not too degenerate, the answer to both questions is Θ_d(|S|/k), where the hidden constant is independent of d, this is clearly best possible up to a multiplicative constant. Analogous results hold in the case of measures. Then we introduce a generalization of the standard covering and packing densities of a convex body C to Borel measure spaces in ℝ^d and, using the aforementioned bounds, we show that they are bounded from above and below, respectively, by functions of d. As an intermediate result, we give a simple proof the existence of weak ϵ-nets of size O(1/ϵ) for the range space induced by all homothets of C. Following some recent work in discrete geometry, we investigate the case d=k=2 in greater detail. We also provide polynomial time algorithms for constructing a packing/covering exhibiting the Θ_d(|S|/k) bound mentioned above in the case that C is an Euclidean ball. Finally, it is shown that if C is a square then it is NP-hard to decide whether S can be covered by |S|/4 squares containing 4 points each.Oriol Solé Piwork_uc5owpu35nbdjhcfxstve5zxtiThu, 17 Nov 2022 00:00:00 GMTCheeger Inequalities for Directed Graphs and Hypergraphs Using Reweighted Eigenvalues
https://scholar.archive.org/work/lj2kaskxd5hqlgoechfrgst2mq
We derive Cheeger inequalities for directed graphs and hypergraphs using the reweighted eigenvalue approach that was recently developed for vertex expansion in undirected graphs [OZ22,KLT22,JPV22]. The goal is to develop a new spectral theory for directed graphs and an alternative spectral theory for hypergraphs. The first main result is a Cheeger inequality relating the vertex expansion ψ⃗(G) of a directed graph G to the vertex-capacitated maximum reweighted second eigenvalue λ⃗_2^v*: λ⃗_2^v*≲ψ⃗(G) ≲√(λ⃗_2^v*·log (Δ/λ⃗_2^v*)). This provides a combinatorial characterization of the fastest mixing time of a directed graph by vertex expansion, and builds a new connection between reweighted eigenvalued, vertex expansion, and fastest mixing time for directed graphs. The second main result is a stronger Cheeger inequality relating the edge conductance ϕ⃗(G) of a directed graph G to the edge-capacitated maximum reweighted second eigenvalue λ⃗_2^e*: λ⃗_2^e*≲ϕ⃗(G) ≲√(λ⃗_2^e*·log (1/λ⃗_2^e*)). This provides a certificate for a directed graph to be an expander and a spectral algorithm to find a sparse cut in a directed graph, playing a similar role as Cheeger's inequality in certifying graph expansion and in the spectral partitioning algorithm for undirected graphs. We also use this reweighted eigenvalue approach to derive the improved Cheeger inequality for directed graphs, and furthermore to derive several Cheeger inequalities for hypergraphs that match and improve the existing results in [Lou15,CLTZ18]. These are supporting results that this provides a unifying approach to lift the spectral theory for undirected graphs to more general settings.Lap Chi Lau, Kam Chuen Tung, Robert Wangwork_lj2kaskxd5hqlgoechfrgst2mqThu, 17 Nov 2022 00:00:00 GMTMetrizable uniform spaces
https://scholar.archive.org/work/h5vzwcuaurfrrjfree4vmotpky
Three themes of general topology: quotient spaces; absolute retracts; and inverse limits - are reapproached here in the setting of metrizable uniform spaces, with an eye to applications in geometric and algebraic topology. The results include: 1) If f: A -> Y is a uniformly continuous map, where X and Y are metric spaces and A is a closed subset of X, we show that the adjunction space X\cup_f Y with the quotient uniformity (hence also with the topology thereof) is metrizable, by an explicit metric. This yields natural constructions of cone, join and mapping cylinder in the category of metrizable uniform spaces, which we show to coincide with those based on subspace (of a normed linear space); on product (with a cone); and on the isotropy of the l_2 metric. 2) We revisit Isbell's theory of uniform ANRs, as refined by Garg and Nhu in the metrizable case. The iterated loop spaces \Omega^n P of a pointed compact polyhedron P are shown to be uniform ANRs. Four characterizations of uniform ANRs among metrizable uniform spaces X are given: (i) the completion of X is a uniform ANR, and the remainder is uniformly a Z-set in the completion; (ii) X is uniformly locally contractible and satisfies the Hahn approximation property; (iii) X is uniformly \epsilon-homotopy dominated by a uniform ANR for each \epsilon>0; (iv) X is an inverse limit of uniform ANRs with "nearly splitting" bonding maps. Several chapters are devoted primarily to exposition: (I) an introduction to uniform spaces, with a focus on the metrizable case; (V) the space of measurable functions; (VI) the space of probability measures and other measure spaces.Sergey A. Melikhovwork_h5vzwcuaurfrrjfree4vmotpkyThu, 17 Nov 2022 00:00:00 GMTTowards combinatorial invariance for Kazhdan-Lusztig polynomials
https://scholar.archive.org/work/o7vb7hdbvnbgdd5wft7rkhfbiu
Kazhdan-Lusztig polynomials are important and mysterious objects in representation theory. Here we present a new formula for their computation for symmetric groups based on the Bruhat graph. Our approach suggests a solution to the combinatorial invariance conjecture for symmetric groups, a well-known conjecture formulated by Lusztig and Dyer in the 1980s.Charles Blundell, Lars Buesing, Alex Davies, Petar Veličković, Geordie Williamsonwork_o7vb7hdbvnbgdd5wft7rkhfbiuWed, 16 Nov 2022 00:00:00 GMTCombinatorially random curves on surfaces
https://scholar.archive.org/work/sw4xdejfgjf4hbfp5urif4lw74
We study topological properties of random closed curves on an orientable surface S of negative Euler characteristic. Letting γ_n denote the conjugacy class of the n^th step of a simple random walk on the Cayley graph driven by a measure whose support is on a finite generating set, then with probability converging to 1 as n goes to infinity, (1) the point in Teichmüller space at which γ_n is length-minimized stays in some compact set; (2) the self-intersection number of γ_n is on the order of n^2, the minimum length of γ_n taken over all hyperbolic metrics is on the order of n, and the metric minimizing the length of γ_n is uniformly thick; and (3) when S is punctured and the distribution is uniform and supported on a generating set of minimum size, the minimum degree of a cover to which γ_n admits a simple elevation (which we call the simple lifting degree of γ_n) grows at least like n/log(n) and at most on the order of n. We also show that these properties are generic, in the sense that the proportion of elements in the ball of radius n in the Cayley graph for which they hold, converges to 1 as n goes to infinity. The lower bounds on simple lifting degree for randomly chosen curves we obtain significantly improve the previously best known bounds which were on the order of log^(1/3)n. As applications, we give relatively sharp upper and lower bounds on the dilatation of a generic point-pushing pseudo-Anosov homeomorphism in terms of the self-intersection number of its defining curve, as well as upper bounds on the simple lifting degree of a random curve in terms of its intersection number which outperform bounds for general curves.Tarik Aougab, Jonah Gasterwork_sw4xdejfgjf4hbfp5urif4lw74Tue, 15 Nov 2022 00:00:00 GMTPair crossing number, cutwidth, and good drawings on arbitrary point sets
https://scholar.archive.org/work/ip3wkan6kjdv3htpbntjv6tdnq
Determining whether there exists a graph such that its crossing number and pair crossing number are distinct is an important open problem in geometric graph theory. We show that cr(G)=O(pcr(G)^3/2) for every graph G, this improves the previous best bound by a logarithmic factor. Answering a question of Pach and Tóth, we prove that the bisection width (and, in fact, the cutwidth as well) of a graph G with degree sequence d_1,d_2,...,d_n satisfies bw(G)=O(√(pcr(G)+∑_k=1^n d_k^2)). Then we show that there is a constant C≥ 1 such that the following holds: For any graph G of order n and any set S of at least n^C points in general position on the plane, G admits a straight-line drawing which maps the vertices to points of S and has no more than O(log n·(pcr(G)+∑_k=1^n d_k^2)) crossings. Our proofs rely on a modified version of a separator theorem for string graphs by Lee, which might be of independent interest.Oriol Solé Piwork_ip3wkan6kjdv3htpbntjv6tdnqTue, 15 Nov 2022 00:00:00 GMTShellability is hard even for balls
https://scholar.archive.org/work/z7vpspyamjgfhpz2hcpshfnku4
The main goal of this paper is to show that shellability is NP-hard for triangulated d-balls (this also gives hardness for triangulated d-manifolds/d-pseudomanifolds with boundary) as soon as d is at least 3. This extends our earlier work with Goaoc, Pat\'akov\'a and Wagner on hardness of shellability of 2-complexes and answers some questions implicitly raised by Danaraj and Klee in 1978 and explicitly mentioned by Santamar\'ia-Galvis and Woodroofe. Together with the main goal, we also prove that collapsibility is NP-hard for 3-complexes embeddable in the 3-space, extending an earlier work of the second author and answering an open question mentioned by Cohen, Fasy, Miller, Nayyeri, Peng and Walkington; and that shellability is NP-hard for 2-complexes embeddable in the 3-space, answering another question of Santamar\'ia-Galvis and Woodroofe (in a slightly stronger form than what is given by the main result).Pavel Paták, Martin Tancerwork_z7vpspyamjgfhpz2hcpshfnku4Tue, 15 Nov 2022 00:00:00 GMTChainlink Polytopes and Ehrhart-Equivalence
https://scholar.archive.org/work/lrchhmlp2jax3psxggsnaekx6a
We introduce a class of polytopes that we call chainlink polytopes and which allow us to construct infinite families of pairs of non isomorphic rational polytopes with the same Ehrhart quasi-polynomial. Our construction is based on circular fence posets, which admit a non-obvious and non-trivial symmetry in their rank sequences that turns out to be reflected in the polytope level. We introduce the related class of chainlink posets and show that they exhibit the same symmetry properties. We further prove an outstanding conjecture on the unimodality of circular rank polynomials.Ezgi Kantarcı Oğuz, Cem Yalım Özel, Mohan Ravichandranwork_lrchhmlp2jax3psxggsnaekx6aTue, 15 Nov 2022 00:00:00 GMTA link condition for simplicial complexes, and CUB spaces
https://scholar.archive.org/work/mwwppx36ofdlvgnjpu432irkqi
We motivate the study of metric spaces with a unique convex geodesic bicombing, which we call CUB spaces. These encompass many classical notions of nonpositive curvature, such as CAT(0) spaces and Busemann-convex spaces. Groups having a geometric action on a CUB space enjoy numerous properties. We want to know when a simplicial complex, endowed with a natural polyhedral metric, is CUB. We establish a link condition, stating essentially that the complex is locally a lattice. This generalizes Gromov's link condition for cube complexes, for the ℓ^∞ metric. The link condition applies to numerous examples, including Euclidean buildings, simplices of groups, Artin complexes of Euclidean Artin groups, (weak) Garside groups, some arcs and curve complexes, and minimal spanning surfaces of knots.Thomas Haettelwork_mwwppx36ofdlvgnjpu432irkqiTue, 15 Nov 2022 00:00:00 GMTOn the power of euclidean division: Lower bounds for algebraic machines, semantically
https://scholar.archive.org/work/hkgtdopqlfa2noluiatrajuft4
This paper presents a new abstract method for proving lower bounds in computational complexity. Based on the notion of topological and measurable entropy for dynamical systems, it is shown to generalise three previous lower bounds results from the literature in algebraic complexity. We use it to prove that 𝚖𝚊𝚡𝚏𝚕𝚘𝚠, a Ptime complete problem, is not computable in polylogarithmic time on parallel random access machines (prams) working with real numbers. This improves on a result of Mulmuley since the class of machines considered extends the class "prams without bit operations". We further improve on this result by showing that euclidean division cannot be computable in polylogarithmic time using division on the reals, pinpointing that euclidean division provides a significant boost in expressive power. On top of showing this new separation result, we show our method captures previous lower bounds results from the literature: Steele and Yao's lower bounds for algebraic decision trees, Ben-Or's lower bounds for algebraic computation trees, Cucker's proof that NC is not equal to Ptime in the real case, and Mulmuley's lower bounds for prams without bit operations.Thomas Seiller and Luc Pellissier and Ulysse Léchinework_hkgtdopqlfa2noluiatrajuft4Tue, 15 Nov 2022 00:00:00 GMT