Abstract. In this paper, we show that any knot group maps onto at most finitely many knot groups. This gives an affirmative answer to a conjecture of J. Simon. We also bound the diameter of a closed hyperbolic 3-manifold linearly in terms of the presentation length of its fundamental group, improving a result of White.

Abstract. In this paper, we show that an aspherical compact graph manifold is nonpositively curved if and only if its fundamental group virtually embeds into a right-angled Artin group. As a consequence, nonpositively curved graph manifolds have linear fundamental groups.

Pythagorean fuzzy sets, as an extension of intuitionistic fuzzy sets to deal with uncertainty, have attracted much attention since their introduction, in both theory and application aspects. In this paper, we investigate multiple attribute decision-making (MADM) problems with Pythagorean linguistic information based on some new aggregation operators. To begin with, we present some new Pythagorean fuzzy linguistic Muirhead mean (PFLMM) operators to deal with MADM problems with Pythagorean fuzzy linguistic information, including the PFLMM operator, the Pythagorean fuzzy linguistic-weighted Muirhead mean operator, the Pythagorean fuzzy linguistic dual Muirhead mean operator and the Pythagorean fuzzy linguistic dual-weighted Muirhead mean operator. The main advantages of these aggregation operators are that they can capture the interrelationships of multiple attributes among any number of attributes by a parameter vector P and make the information aggregation process more flexible by the parameter vector P. In addition, some of the properties of these new aggregation operators are proved and some special cases are discussed where the parameter vector takes some different values. Moreover, we present two new methods to solve MADM problems with Pythagorean fuzzy linguistic information. Finally, an illustrative example is provided to show the feasibility and validity of the new methods, to investigate the influences of parameter vector P on decision-making results, and also to analyze the advantages of the proposed methods by comparing them with the other existing methods. K E Y W O R D SMuirhead mean operators, multiattribute decision making, Pythagorean fuzzy set

We study the set vol(M, G) of volumes of all representations ρ : π1M → G, where M is a closed oriented 3-manifold and G is either Iso+H 3 or Isoe SL2(R).By various methods, including relations between the volume of representations and the Chern-Simons invariants of flat connections, and recent results of surfaces in 3-manifolds, we prove that any 3-manifold M with positive Gromov simplicial volume has a finite cover M with vol( M, Iso+H 3 ) = {0}, and that any non-geometric 3-manifold M containing at least one Seifert piece has a finite cover M with vol( M, Isoe SL2(R)) = {0}.We also find 3-manifolds M with positive simplicial volume but vol(M, Iso+H 3 ) = {0}, and non-trivial graph manifolds M with vol(M, Isoe SL2(R)) = {0}, proving that it is in general necessary to pass to some finite covering to guarantee that vol(M, G) = {0}.Besides we determine vol(M, G) when M supports the Seifert geometry. ContentsDefinition 4.9. Let M be an orientable closed irreducible mixed 3-manifold containing no essential Klein bottles. Let J 0 be a JSJ piece, T 0 be a JSJ torus adjacent to J 0 and ζ 0 be a slope on T 0 . A partial PW subsurface j : R M is said to be virtually bounded by ζ 0 outside J 0 if the boundary ∂R of R is non-empty, covering ζ 0 under j, and if the interiorR of R misses J 0 under j. In this case, the carrier chunk X(R) ⊂ M of R is the unique minimal chunk that contains R, and the carrier boundary of X(R) is the component T 0 ⊂ ∂X(R).Definition 4.10. We say that a partial PW subsurface j : R M is parallel cutting if, for every JSJ torus T ⊂ M , all components of j −1 (T ) in R cover the same slope of T . Virtual existence of partial PW subsurfacesTheorem 4.11. Let M be an orientable closed irreducible mixed 3-manifold containing no essential Klein bottles. Let ζ 0 be a slope on a JSJ torus T 0 adjacent to a JSJ piece J 0 . Then, for some finite coverM of M together with an elevation (J 0 ,T 0 ,ζ 0 ) of the triple (J 0 , T 0 , ζ 0 ), Pierre Derbez I2M UMR 7373

This essay designs an innovate approach to work out linguistic multiattribute group decision‐making (MAGDM) issues with complex q‐rung orthopair fuzzy 2‐tuple linguistic (Cq‐ROF2TL) evaluation information. To begin with, the conception of Cq‐ROF2TL set is propounded to express uncertain and fuzzy assessment information. Meanwhile, the score and accuracy function, a comparison approach, Cq‐ROF2TL weighted averaging, and Cq‐ROF2TL weighted geometric operator are put forward. Furthermore, to take into consideration the correlation among multiple input data, the Cq‐ROF2TL Maclaurin symmetric mean (MSM) operator, the Cq‐ROF2TL dual MSM operator and their weighted forms are presented. Several attractive characteristics and particular instances of the developed operators are also explored at length. Later, an innovative MAGDM methodology is designed based upon the propounded operators to settle the emergency program evaluation issue under the Cq‐ROF2TL circumstance. Consequently, the efficiency and outstanding superiority of the created approach are severally substantiated by parameter exploration and detailed comparative analysis.

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Linguistic Pythagorean fuzzy (LPF) set is an efficacious technique to comprehensively represent uncertain assessment information by combining the Pythagorean fuzzy numbers and linguistic variables. In this paper, we define several novel essential operations of LPF numbers based upon Einstein operations and discuss several relations between these operations. For solving the LPF numbers fusion problem, several LPF aggregation operators, including LPF Einstein weighted averaging (LPFEWA) operator, LPF Einstein weighted geometric (LPFEWG) operator and LPF Einstein hybrid operator, are propounded; the prominent characteristics of these operators are investigated as well. Furthermore, a multi-attribute group decision making (MAGDM) approach is presented on the basis of the developed operators under an LPF environment. Ultimately, two application cases are utilized to demonstrate the practicality and feasibility of the developed decision approach and the comparison analysis is provided to manifest the merits of it.

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