Journal of Topology and Analysis

The goal of the present work is to introduce new metric space techniques to study the intersection properties of families of balls. These new techniques add, on the one hand, to results due to Lindenstrauss on the extension of uniformly continuous functions and compact linear operators, and answer, on the other hand, questions raised by Aronszajn and Panitchpakdi on hyperconvex metric spaces. The present work is divided into two parts.

In the first part, the proofs of our two main results on ball intersection properties in metric spaces are presented. The first main result states that for any integer n greater or equal to three, a complete and almost n-hyperconvex metric space is automatically n-hyperconvex. The second main result shows that in complete metric spaces, the property of being 4-hyperconvex is equivalent to the property of being finitely hyperconvex and that the analogues are true for externally 4-hyperconvex and weakly externally 4-hyperconvex subsets. This last result and the results proved later in this work unify the analysis of those three properties: hyperconvexity, external hyperconvexity and weak external hyperconvexity.

In the second part of this work, we make the link with notions of convexity and bicombings and present applications. We extend local-to-global results on hyperconvex spaces to externally hyperconvex spaces as well as to weakly externally hyperconvex spaces. We conclude with applications of our results to the characterization of externally hyperconvex and weakly externally hyperconvex subsets of hyperconvex spaces.

In this paper, we consider the Galois covers of algebraic surfaces of degree 6, with all associated planar degenerations. We compute the fundamental groups of those Galois covers, using their degeneration. We show that for 8 types of degenerations, the fundamental group of the Galois cover is non-trivial and for 20 types it is trivial. Moreover, we compute the Chern numbers of all the surfaces with this type of degeneration and prove that the signatures of all their Galois covers are negative. We formulate a conjecture regarding the structure of the fundamental groups of the Galois covers based on our findings.

By a Morse function on a compact manifold with boundary we mean a real-valued function without critical points near the boundary such that its critical points as well as the critical points of its restriction to the boundary are all nondegenerate. For such Morse functions, Saeki and Yamamoto have previously defined a certain notion of cusp cobordism, and computed the unoriented cusp cobordism group of Morse functions on surfaces. In this paper, we compute unoriented and oriented cusp cobordism groups of Morse functions on manifolds of any dimension by employing Levine’s cusp elimination technique as well as the complementary process of creating pairs of cusps along fold lines. We show that both groups are cyclic of order two in even dimensions, and cyclic of infinite order in odd dimensions. For Morse functions on surfaces our result yields an explicit description of Saeki–Yamamoto’s cobordism invariant which they constructed by means of the cohomology of the universal complex of singular fibers.

We give some local moves of the Stein factorization of the product map of two Morse functions on a closed orientable smooth 3-manifold which can be realized by isotopies of the functions.

We prove uniqueness, up to diffeomorphism, of symplectically aspherical fillings of certain unit cotangent bundles, including those of higher-dimensional tori.

A group $G$ is called bounded if every conjugation-invariant norm on $G$ has finite diameter. We introduce various strengthenings of this property and investigate them in several classes of groups including semisimple Lie groups, arithmetic groups and linear algebraic groups. We provide applications to Hamiltonian dynamics.

We study the “geometric Ricci curvature lower bound”, introduced previously by Junge, Li and LaRacuente, for a variety of examples including group von Neumann algebras, free orthogonal quantum groups $O_N^{+}$, $q$-deformed Gaussian algebras and quantum tori. In particular, we show that Laplace operator on $O_N^{+}$ admits a factorization through the Laplace–Beltrami operator on the classical orthogonal group, which establishes the first connection between these two operators. Based on a non-negative curvature condition, we obtain the completely bounded version of the modified log-Sobolev inequalities for the corresponding quantum Markov semigroups on the examples mentioned above. We also prove that the “geometric Ricci curvature lower bound” is stable under tensor products and amalgamated free products. As an application, we obtain a sharp Ricci curvature lower bound for word-length semigroups on free group factors.

The existence of Kähler Einstein metrics with mixed cone and cusp singularity has received considerable attentions in recent years. It is believed that such kind of metric would give rise to important geometric invariants. We computed their $L^2$-Hodge–Frölicher spectral sequence under the Dirichlet and Neumann boundary conditions and examine the pure Hodge structures on them. It turns out that these cohomologies agree well with the de Rham cohomology of a good compactification.

The Wiman–Edge pencil is a pencil of genus 6 curves for which the generic member has automorphism group the alternating group $A_5$. There is a unique smooth member, the Wiman sextic, with automorphism group the symmetric group $S_5$. Farb and Looijenga proved that the monodromy of the Wiman–Edge pencil is commensurable with the Hilbert modular group ${\text{SL}}_2(\mathbb{Z}[\sqrt{5}])$. In this note, we give a complete description of the monodromy by congruence conditions modulo 4 and 5. The congruence condition modulo 4 is new, and this answers a question of Farb–Looijenga. We also show that the smooth resolution of the Baily–Borel compactification of the locally symmetric manifold associated with the monodromy is a projective surface of general type. Lastly, we give new information about the image of the period map for the pencil.

A group is SimpHAtic if it acts geometrically on a simply connected simplicially hereditarily aspherical (SimpHAtic) complex. We show that finitely presented normal subgroups of the SimpHAtic groups are either:

• finite, or
• of finite index, or
• virtually free.

This result applies, in particular, to normal subgroups of systolic groups. We prove similar strong restrictions on group extensions for other classes of asymptotically aspherical groups.

The proof relies on studying homotopy types at infinity of groups in question.

We also show that non-uniform lattices in SimpHAtic complexes (and in more general complexes) are not finitely presentable and that finitely presented groups acting properly on such complexes act geometrically on SimpHAtic complexes. In Appendix we present the topological two-dimensional quasi-Helly property of systolic complexes.