The universal abelian cover of a graph manifold

2013 ◽  
Author(s):  
Helge Pedersen
Keyword(s):  
Abelian Cover ◽  
Graph Manifold

scholarly journals Conjugacy Problem in the Fundamental Groups of High-Dimensional Graph Manifolds

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1330
Author(s):  
Raeyong Kim
Keyword(s):  
Group Structure ◽  
Conjugacy Problem ◽  
High Dimensional ◽  
Fundamental Groups ◽  
Graph Manifold ◽  
Solvable Word Problem ◽  
Algorithmic Problems ◽  
Graph Manifolds ◽  
Dimensional Graph ◽  
Solvable Word

The conjugacy problem for a group G is one of the important algorithmic problems deciding whether or not two elements in G are conjugate to each other. In this paper, we analyze the graph of group structure for the fundamental group of a high-dimensional graph manifold and study the conjugacy problem. We also provide a new proof for the solvable word problem.

scholarly journals The Asymptotics of the Higher Dimensional Reidemeister Torsion for Exceptional Surgeries Along Twist Knots

2018 ◽  
Vol 61 (1) ◽  
pp. 211-224 ◽  
Author(s):  
Anh T. Tran ◽  
Yoshikazu Yamaguchi
Keyword(s):  
Asymptotic Behavior ◽  
Reidemeister Torsion ◽  
Graph Manifold ◽  
Higher Dimensional ◽  
Graph Manifolds

AbstractWe determine the asymptotic behavior of the higher dimensional Reidemeister torsion for the graph manifolds obtained by exceptional surgeries along twist knots. We show that all irreducible SL2()-representations of the graph manifold are induced by irreducible metabelian representations of the twist knot group. We also give the set of the limits of the leading coeõcients in the higher dimensional Reidemeister torsion explicitly.

scholarly journals Optimal results in Lorentzian Aubry–Mather theory

2020 ◽  
Author(s):  
Stefan Suhr
Keyword(s):  
Fixed Point ◽  
Invariant Measures ◽  
Lipschitz Continuity ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Multiplicity Results ◽  
Time Separation ◽  
Abelian Cover ◽  
Mather Theory ◽  
Maximal Invariant

AbstractThis article complements the Lorentzian Aubry–Mather Theory in Suhr (Geom Dedicata 160:91–117, 2012; J Fixed Point Theory Appl 21:71, 2019) by giving optimal multiplicity results for the number of maximal invariant measures. As an application the optimal Lipschitz continuity of the time separation on the Abelian cover is established.

scholarly journals Local Mixing and Invariant Measures for Horospherical Subgroups on Abelian Covers

2018 ◽  
Vol 2019 (19) ◽  
pp. 6036-6088
Author(s):  
Hee Oh ◽  
Wenyu Pan
Keyword(s):  
Lie Group ◽  
Invariant Measures ◽  
Dense Subgroup ◽  
Abelian Cover ◽  
Rank One ◽  
Hyperbolic Three Manifolds ◽  
Abelian Covers ◽  
Prime Geodesic Theorem ◽  
Convex Cocompact ◽  
Frame Flow

Abstract Abelian covers of hyperbolic three-manifolds are ubiquitous. We prove the local mixing theorem of the frame flow for abelian covers of closed hyperbolic three-manifolds. We obtain a classification theorem for measures invariant under the horospherical subgroup. We also describe applications to the prime geodesic theorem as well as to other counting and equidistribution problems. Our results are proved for any abelian cover of a homogeneous space Γ0∖G where G is a rank one simple Lie group and Γ0 < G is a convex cocompact Zariski dense subgroup.

scholarly journals L-spaces, taut foliations, and graph manifolds

2020 ◽  
Vol 156 (3) ◽  
pp. 604-612 ◽  
Author(s):  
Jonathan Hanselman ◽  
Jacob Rasmussen ◽  
Sarah Dean Rasmussen ◽  
Liam Watson
Keyword(s):  
Graph Manifold ◽  
Graph Manifolds

If $Y$ is a closed orientable graph manifold, we show that $Y$ admits a coorientable taut foliation if and only if $Y$ is not an L-space. Combined with previous work of Boyer and Clay, this implies that $Y$ is an L-space if and only if $\unicode[STIX]{x1D70B}_{1}(Y)$ is not left-orderable.

The Alexander module of a knotted theta-curve

1989 ◽  
Vol 106 (1) ◽  
pp. 95-106 ◽  
Author(s):  
Rick Litherland
Keyword(s):  
Group Ring ◽  
Torsion Module ◽  
Abelian Cover

Let K be a knotted theta-curve with exterior X, and let ∂_X be one of the two pieces into which ∂X is divided by the meridians of the edges of K. Let X be the universal abelian cover of X. Then is a module over the group ring of H1(X); i.e. over . We call this the Alexander module of K, and denote it by A(K). This, rather than H1(X), seems to be the analogue of the Alexander module of a classical knot; it is a torsion module of deficiency 0. Moreover, it is not an invariant of X alone.

Graph manifold ℤ-homology 3-spheres and taut foliations

2015 ◽  
Vol 8 (2) ◽  
pp. 571-585 ◽  
Author(s):  
Michel Boileau ◽  
Steven Boyer
Keyword(s):  
Graph Manifold

ON THE COMPUTATION OF THE TURAEV-VIRO MODULE OF A KNOT

1998 ◽  
Vol 07 (07) ◽  
pp. 843-856 ◽  
Author(s):  
H. ABCHIR ◽  
C. BLANCHET
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Fundamental Domain ◽  
Topological Quantum Field Theory ◽  
Quantum Field ◽  
Abelian Cover ◽  
Topological Quantum ◽  
Special Case

Let M be the manifold obtained by 0-framed surgery along a knot K in the 3-sphere. A Topological Quantum Field Theory assigns to a fundamental domain of the universal abelian cover of M an operator, whose non-nilpotent part is the Turaev-Viro module of K. In this paper, using surgery formulas, we give a matrix presentation for the Turaev-Viro module of any knot K, in the case of the (Vp, Zp) TQFT of Blanchet, Habegger, Masbaum and Vogel. We do the computation for a family of knots in the special case p = 8, and note the relation with the fibering question.

scholarly journals Non-normal abelian covers

2012 ◽  
Vol 148 (4) ◽  
pp. 1051-1084 ◽  
Author(s):  
Valery Alexeev ◽  
Rita Pardini
Keyword(s):  
Abelian Group ◽  
Moduli Space ◽  
Moduli Spaces ◽  
General Type ◽  
Finite Abelian Group ◽  
Algebraic Varieties ◽  
Surfaces Of General Type ◽  
Normal Case ◽  
Abelian Cover ◽  
Abelian Covers

AbstractAn abelian cover is a finite morphism X→Y of varieties which is the quotient map for a generically faithful action of a finite abelian group G. Abelian covers with Y smooth and X normal were studied in [R. Pardini, Abelian covers of algebraic varieties, J. Reine Angew. Math. 417 (1991), 191–213; MR 1103912(92g:14012)]. Here we study the non-normal case, assuming that X and Y are S2 varieties that have at worst normal crossings outside a subset of codimension greater than or equal to two. Special attention is paid to the case of ℤr2-covers of surfaces, which is used in [V. Alexeev and R. Pardini, Explicit compactifications of moduli spaces of Campedelli and Burniat surfaces, Preprint (2009), math.AG/arXiv:0901.4431] to construct explicitly compactifications of some components of the moduli space of surfaces of general type.

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