Isomorphism classification of Leary–Minasyan groups

Recently, I. J. Leary and A. Minasyan [Commensurating HNN extensions: Nonpositive curvature and biautomaticity, Geom. Topol. 25 (2021), 4, 1819–1860] studied the class of groups G ⁢ ( A , L ) G(A,L) defined as commensurating HNN-extensions of Z n \mathbb{Z}^{n} . This class, containing the class of Baumslag–Solitar groups, also includes other groups with curious properties, such as being CAT(0) but not biautomatic. In this paper, we classify the groups G ⁢ ( A , L ) G(A,L) up to isomorphism.

$$\alpha $$-Dirac-harmonic maps from closed surfaces

Abstract$$\alpha $$ α -Dirac-harmonic maps are variations of Dirac-harmonic maps, analogous to $$\alpha $$ α -harmonic maps that were introduced by Sacks–Uhlenbeck to attack the existence problem for harmonic maps from closed surfaces. For $$\alpha >1$$ α > 1 , the latter are known to satisfy a Palais–Smale condition, and so, the technique of Sacks–Uhlenbeck consists in constructing $$\alpha $$ α -harmonic maps for $$\alpha >1$$ α > 1 and then letting $$\alpha \rightarrow 1$$ α → 1 . The extension of this scheme to Dirac-harmonic maps meets with several difficulties, and in this paper, we start attacking those. We first prove the existence of nontrivial perturbed $$\alpha $$ α -Dirac-harmonic maps when the target manifold has nonpositive curvature. The regularity theorem then shows that they are actually smooth if the perturbation function is smooth. By $$\varepsilon $$ ε -regularity and suitable perturbations, we can then show that such a sequence of perturbed $$\alpha $$ α -Dirac-harmonic maps converges to a smooth coupled $$\alpha $$ α -Dirac-harmonic map.

Improved generalized periods estimates over curves on Riemannian surfaces with nonpositive curvature

We show that, on compact Riemannian surfaces of nonpositive curvature, the generalized periods, i.e. the 𝜈-th order Fourier coefficients of eigenfunctions e λ e_{\lambda} over a closed smooth curve 𝛾 which satisfies a natural curvature condition, go to 0 at the rate of O ⁢ ( ( log ⁡ λ ) - 1 2 ) O((\log\lambda)^{-\frac{1}{2}}) in the high energy limit λ → ∞ \lambda\to\infty if 0 < | ν | λ < 1 - δ 0<\frac{\lvert\nu\rvert}{\lambda}<1-\delta for any fixed 0 < δ < 1 0<\delta<1 . Our result implies, for instance, that the generalized periods over geodesic circles on any surfaces with nonpositive curvature would converge to zero at the rate of O ⁢ ( ( log ⁡ λ ) - 1 2 ) O((\log\lambda)^{-\frac{1}{2}}) .

A Mean Ergodic Theorem for Affine Nonexpansive Mappings in Nonpositive Curvature Metric Spaces

In this paper, we consider the orbits of an affine nonexpansive mapping in Hadamard (nonpositive curvature metric) spaces and prove an ergodic theorem for the inductive mean, which extends the von Neumann linear ergodic theorem. The main result shows that the sequence given by the inductive means of iterations of an affine nonexpansive mapping with a nonempty fixed point set converges strongly to a fixed point of the mapping. A Tauberian theorem is also proved in order to ensure convergence of the iterations.

Helly meets Garside and Artin

A graph is Helly if every family of pairwise intersecting combinatorial balls has a nonempty intersection. We show that weak Garside groups of finite type and FC-type Artin groups are Helly, that is, they act geometrically on Helly graphs. In particular, such groups act geometrically on spaces with a convex geodesic bicombing, equipping them with a nonpositive-curvature-like structure. That structure has many properties of a CAT(0) structure and, additionally, it has a combinatorial flavor implying biautomaticity. As immediate consequences we obtain new results for FC-type Artin groups (in particular braid groups and spherical Artin groups) and weak Garside groups, including e.g. fundamental groups of the complements of complexified finite simplicial arrangements of hyperplanes, braid groups of well-generated complex reflection groups, and one-relator groups with non-trivial center. Among the results are: biautomaticity, existence of EZ and Tits boundaries, the Farrell–Jones conjecture, the coarse Baum–Connes conjecture, and a description of higher order homological and homotopical Dehn functions. As a means of proving the Helly property we introduce and use the notion of a (generalized) cell Helly complex.