Geometriae Dedicata最新文献

Let (X=X_1times X_2) be a product of two rank one symmetric spaces of non-compact type and (Gamma ) a torsion-free discrete subgroup in (G_1times G_2). We show that the spectrum of (Gamma backslash (X_1times 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 ).

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.

Consider a transitive Anosov flow on a closed 3-manifold. After removing a finite set of null-homologous periodic orbits, we study the distribution of the remaining periodic orbits in the homology of the knot complement.

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 quasi-isometry 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.

In this note, we prove a well-known 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).

We will prove that any non-empty open set in every complete connected metric space (X, d), where balls have compact closures, contains a paradoxical (uncountable) set relative to a non-supramenable connected Lie group that acts continuously and transitively on X.

Let S be a (P^2)-knot which is the connected sum of a 2-knot with normal Euler number 0 and an unknotted (P^2)-knot with normal Euler number ({pm }{2}) in a closed 4-manifold 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 boundary-stabilizations 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 4-dimensional analogue of Waldhausen’s theorem on Heegaard splittings.

In this paper we use the Weitzenböck formulas to get information about the Betti numbers of compact nearly (G_2) and compact nearly Kähler 6-manifolds. First, we establish estimates on two curvature-type 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.

For the real symplectic groups (G=textrm{Sp}(n,mathbb {R})), we classify all the Klein four-symmetric pairs ((G,G^Gamma )), and determine whether there exist infinite-dimensional 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).

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.