Geometriae Dedicata
Journal Prestige (SJR): 1.255 Citation Impact (citeScore): 1 Number of Followers: 3 Hybrid journal (It can contain Open Access articles) ISSN (Print) 15729168  ISSN (Online) 00465755 Published by SpringerVerlag [2468 journals] 
 Top degree $$\ell ^p$$ homology and conformal dimension of buildings

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Abstract: Abstract For a noncompact finite thickness building whose Davis apartment is an orientable pseudomanifold, we compute the supremum of the set of \(p>1\) such that its top dimensional reduced \(\ell ^p\) cohomology is nonzero. We adapt the nonvanishing assertion of this result to any finite thickness building using the Bestvina realization. Using similar techniques, we generalize bounds obtained by Clais on the conformal dimension of some Gromovhyperbolic buildings to any such building.
PubDate: 20240523

 Chern–Simons theory and cohomological invariants of representation
varieties
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Abstract: Abstract We prove a general local rigidity theorem for pullbacks of homogeneous forms on reductive symmetric spaces under representations of discrete groups. One application of the theorem is that the volume of a closed manifold locally modelled on a reductive homogeneous space G/H is constant under deformation of the G/Hstructure. The proof elaborates on an argument given by Labourie for closed antide Sitter 3manifolds. The core of the work is a reinterpretation of old results of Cartan, Chevalley and Borel, showing that the algebra of Ginvariant forms on G/H is generated by “Chern–Weil forms” and “Chern–Simons forms”.
PubDate: 20240520

 Isoperiodic families of Poncelet polygons inscribed in a circle and
circumscribed about conics from a confocal pencil
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Abstract: Abstract Poncelet polygons inscribed in a circle and circumscribed about conics from a confocal family naturally arise in the analysis of the numerical range and Blaschke products. We examine the behaviour of such polygons when the inscribed conic varies through a confocal pencil and discover cases when each conic from the confocal family is inscribed in an npolygon, which is inscribed in the circle, with the same n. Complete geometric characterization of such cases for \(n\in \{4,6\}\) is given and proved that this cannot happen for other values of n. We establish a relationship of such families of Poncelet quadrangles and hexagons to solutions of a Painlevé VI equation.
PubDate: 20240514

 Fundamental groups and group presentations with bounded relator lengths

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Abstract: Abstract We study the geometry of compact geodesic spaces with trivial first Betti number admitting large finite groups of isometries. We show that if a finite group G acts by isometries on a compact geodesic space X whose first Betti number vanishes, then \({\text {diam}}(X) / {\text {diam}}(X / G ) \le 4 \sqrt{ \vert G \vert }\) . For a group G and a finite symmetric generating set S, \(P_k(\varGamma (G, S))\) denotes the 2dimensional CWcomplex whose 1skeleton is the Cayley graph \(\varGamma \) of G with respect to S and whose 2cells are mgons for \(0 \le m \le k\) , defined by the simple graph loops of length m in \(\varGamma \) , up to cyclic permutations. Let G be a finite abelian group with \(\vert G \vert \ge 3\) and S a symmetric set of generators for which \(P_k(\varGamma (G,S))\) has trivial first Betti number. We show that the first nontrivial eigenvalue \(\lambda _1\) of the Laplacian on the Cayley graph satisfies \(\lambda _1 \ge 2  2 \cos ( 2 \pi / k ) \) . We also give an explicit upper bound on the diameter of the Cayley graph of G with respect to S of the form \(O (k^2 \vert S \vert \log \vert G \vert )\) . Related explicit bounds for the Cheeger constant and Kazhdan constant of the pair (G, S) are also obtained.
PubDate: 20240510

 The Liouville current of holomorphic quadratic differential metrics

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Abstract: Abstract In this paper we study the Liouville current of flat cone metrics coming from a holomorphic quadratic differential. Anja Bankovic and Christopher J. Leininger proved in Bankovic and Leininger (Trans Am Math Soc 370:1867–1884, 2018) that for a fixed closed surface, there is an injection map from the space of flat cone metrics to the space of geodesic currents. We manage to show that metrics coming from holomorphic quadratic differentials can be distinguished from other flat metrics by just looking at the geodesic currents. The key idea is to analyze the support of Liouville current, which is a topological invariant independent of the metric, and get information about cone angles and holonomy. The holonomy part involves some subtlety of relationship between singular foliation and geodesic lamination. We also obtain a new proof of a classical result that almost all simple geodesics of a quadratic differential metric will be dense in the surface. Furthermore, for other flat cone metrics, there is no simple dense geodesic.
PubDate: 20240507

 An extension of the Thurston metric to projective filling currents

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Abstract: Abstract We study the geometry of the space of projectivized filling geodesic currents \(\mathbb {P}\mathcal {C}_{fill}(S)\) . Bonahon showed that Teichmüller space, \(\mathcal {T}(S)\) embeds into \(\mathbb {P}\mathcal {C}_{fill}(S)\) . We extend the symmetrized Thurston metric from \(\mathcal {T}(S)\) to the entire (projectivized) space of filling currents, and we show that \(\mathcal {T}(S)\) is isometrically embedded into the bigger space. Moreover, we show that there is no quasiisometric projection back down to \(\mathcal {T}(S)\) . Lastly, we study the geometry of a lengthminimizing projection from \(\mathbb {P}\mathcal {C}_{fill}(S)\) to \(\mathcal {T}(S)\) defined previously by Hensel and the author.
PubDate: 20240506

 Temperedness of locally symmetric spaces: the product case

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Abstract: Abstract Let \(X=X_1\times X_2\) be a product of two rank one symmetric spaces of noncompact type and \(\Gamma \) a torsionfree discrete subgroup in \(G_1\times G_2\) . We show that the spectrum of \(\Gamma \backslash (X_1\times X_2)\) is related to the asymptotic growth of \(\Gamma \) in the two directions defined by the two factors. We obtain that \(L^2(\Gamma \backslash (G_1 \times G_2))\) is tempered for a large class of \(\Gamma \) .
PubDate: 20240503

 Minimality and unique ergodicity of Veech 1969 type interval exchange
transformations
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Abstract: Abstract We give conditions for minimality of \({\mathbb {Z}}/N{\mathbb {Z}}\) extensions of a rotation of angle \(\alpha \) with one marked point, solving the problem for any prime N: for \(N=2\) , these correspond to the Veech 1969 examples, for which a necessary and sufficient condition was not known yet. We provide also a word combinatorial criterion of minimality valid for general interval exchange transformations, which applies to \({\mathbb {Z}}/N{\mathbb {Z}}\) extensions of any interval exchange transformation with any number of marked points. Then we give a condition for unique ergodicity of these extensions when the initial interval exchange transformation is linearly recurrent and there are one or two marked points.
PubDate: 20240503

 Distribution of periodic orbits in the homology group of a knot complement

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Abstract: Abstract Consider a transitive Anosov flow on a closed 3manifold. After removing a finite set of nullhomologous periodic orbits, we study the distribution of the remaining periodic orbits in the homology of the knot complement.
PubDate: 20240429

 Decomposition complexity growth of finitely generated groups

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Abstract: Abstract Finite decomposition complexity and asymptotic dimension growth are two generalizations of M. Gromov’s asymptotic dimension which can be used to prove property A for large classes of finitely generated groups of infinite asymptotic dimension. In this paper, we introduce the notion of decomposition complexity growth, which is a quasiisometry invariant generalizing both finite decomposition complexity and dimension growth. We show that subexponential decomposition complexity growth implies property A, and is preserved by certain group and metric constructions.
PubDate: 20240429

 Scalar curvature along the Ricci flow

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Abstract: Abstract In this note, we prove a wellknown conjecture on the Ricci flow under a curvature condition, which is a pinching between the Ricci and Weyl tensors divided by suitably translated scalar curvature, motivated by Cao’s result (Commun Anal Geom 19(5):975–990, 2011).
PubDate: 20240415

 On bounded paradoxical sets and Lie groups

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Abstract: Abstract We will prove that any nonempty open set in every complete connected metric space (X, d), where balls have compact closures, contains a paradoxical (uncountable) set relative to a nonsupramenable connected Lie group that acts continuously and transitively on X.
PubDate: 20240414

 Trisections obtained by trivially regluing surfaceknots

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Abstract: Abstract Let S be a \(P^2\) knot which is the connected sum of a 2knot with normal Euler number 0 and an unknotted \(P^2\) knot with normal Euler number \({\pm }{2}\) in a closed 4manifold X with trisection \(T_{X}\) . Then, we show that the trisection of X obtained by the trivial gluing of relative trisections of \(\overline{\nu (S)}\) and \(X\nu (S)\) is diffeomorphic to a stabilization of \(T_{X}\) . It should be noted that this result is not obvious since boundarystabilizations introduced by Kim and Miller are used to construct a relative trisection of \(X\nu (S)\) . As a corollary, if \(X=S^4\) and \(T_X\) was the genus 0 trisection of \(S^4\) , the resulting trisection is diffeomorphic to a stabilization of the genus 0 trisection of \(S^4\) . This result is related to the conjecture that is a 4dimensional analogue of Waldhausen’s theorem on Heegaard splittings.
PubDate: 20240413

 Betti numbers of nearly $$G_2$$ and nearly Kähler 6manifolds with
Weyl curvature bounds
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Abstract: Abstract In this paper we use the Weitzenböck formulas to get information about the Betti numbers of compact nearly \(G_2\) and compact nearly Kähler 6manifolds. First, we establish estimates on two curvaturetype self adjoint operators on particular spaces assuming bounds on the sectional curvature. Then using the Weitzenböck formulas on harmonic forms, we get results of the form: if certain lower bounds hold for these curvature operators then certain Betti numbers are zero. Finally, we combine both steps above to get sufficient conditions of vanishing of certain Betti numbers based on the bounds on the sectional curvature.
PubDate: 20240412

 Branching laws of Klein foursymmetric pairs for $$\textrm{Sp}(n,\mathbb
{R})$$
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Abstract: Abstract For the real symplectic groups \(G=\textrm{Sp}(n,\mathbb {R})\) , we classify all the Klein foursymmetric pairs \((G,G^\Gamma )\) , and determine whether there exist infinitedimensional irreducible \((\mathfrak {g},K)\) modules discretely decomposable upon restriction to \(G^\Gamma \) . As a consequence, we obtain a similar result to Chen and He (Int J Math 34(1):2250094, 2023, Corollary 21).
PubDate: 20240411

 Topological aspects of the space of metric measure spaces

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Abstract: Abstract Gromov introduced two distance functions, the box distance and the observable distance, on the space of isomorphism classes of metric measure spaces and developed the convergence theory of metric measure spaces. We investigate several topological properties on the space equipped with these distance functions toward a deep understanding of convergence theory.
PubDate: 20240411

 The Nielsen realization problem for high degree del Pezzo surfaces

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Abstract: Abstract Let M be a smooth 4manifold underlying some del Pezzo surface of degree \(d \ge 6\) . We consider the smooth Nielsen realization problem for M: which finite subgroups of \({{\,\textrm{Mod}\,}}(M) = \pi _0({{\,\textrm{Homeo}\,}}^+(M))\) have lifts to \({{\,\textrm{Diff}\,}}^+(M) \le {{\,\textrm{Homeo}\,}}^+(M)\) under the quotient map \(\pi : {{\,\textrm{Homeo}\,}}^+(M) \rightarrow {{\,\textrm{Mod}\,}}(M)\) ' We give a complete classification of such finite subgroups of \({{\,\textrm{Mod}\,}}(M)\) for \(d \ge 7\) and a partial answer for \(d = 6\) . For the cases \(d \ge 8\) , the quotient map \(\pi \) admits a section with image contained in \({{\,\textrm{Diff}\,}}^+(M)\) . For the case \(d = 7\) , we show that all finite order elements of \({{\,\textrm{Mod}\,}}(M)\) have lifts to \({{\,\textrm{Diff}\,}}^+(M)\) , but there are finite subgroups of \({{\,\textrm{Mod}\,}}(M)\) that do not lift to \({{\,\textrm{Diff}\,}}^+(M)\) . We prove that the condition of whether a finite subgroup \(G \le {{\,\textrm{Mod}\,}}(M)\) lifts to \({{\,\textrm{Diff}\,}}^+(M)\) is equivalent to the existence of a certain equivariant connected sum realizing G. For the case \(d = 6\) , we show this equivalence for all maximal finite subgroups \(G \le {{\,\textrm{Mod}\,}}(M)\) .
PubDate: 20240408

 On the Teichmüller stack of compact quotients of $${\text
{SL}}_2({\mathbb {C}})$$
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Abstract: Abstract This article aims to pursue and generalize, by using the global point of view offered by the stacks, the local study made by Ghys (J für die reine und angewandte Mathematik 468:113–138, 1995) concerning the deformations of complex structures of compact quotients of \({\text {SL}}_2({\mathbb {C}})\) . In his article, Ghys showed that the analytic germ of the representation variety \({\mathcal {R}}(\varGamma ):={\text {Hom}}(\varGamma ,{\text {SL}}_2({\mathbb {C}}))\) of \(\varGamma \) in \({\text {SL}}_2({\mathbb {C}})\) , pointed at the trivial morphism, determines the Kuranishi space of \({\text {SL}}_2({\mathbb {C}})/\varGamma \) . In this note, we show that the tautological family above a Zariski analytic open subset V in \({\mathcal {R}}(\varGamma )\) remains complete. Moreover, the computation of the isotropy group of a complex structure in Teichmüller space, allows us to affirm that the quotient stack \([V/{\text {SL}}_2({\mathbb {C}})]\) is an open substack of the Teichmüller stack of \({\text {SL}}_2({\mathbb {C}})/\varGamma \) .
PubDate: 20240404

 Correction: Intersection theory and volumes of moduli spaces of flat
metrics on the sphere
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PubDate: 20240403
DOI: 10.1007/s1071102400909z

 Variation of holonomy for projective structures and an application to
drilling hyperbolic 3manifolds
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Abstract: Abstract We bound the derivative of complex length of a geodesic under variation of the projective structure on a closed surface in terms of the norm of the Schwarzian in a neighborhood of the geodesic. One application is to conemanifold deformations of acylindrical hyperbolic 3manifolds.
PubDate: 20240403
DOI: 10.1007/s10711024009080
