arXiv research

Two groups with same profinite completion have different co-Hopfian properties.

Study big mapping class groups and their co-Hopfian property, finding new examples and proving injective homomorphisms results.

For all but finitely many compact orientable surfaces, we show that any superinjective map from the complex of separating curves into itself is induced by an element of the extended mapping class group. We apply this result to proving that any finite index subgroup of the Johnson kernel is co-Hopfian. Analogous propert…

Let g and n be integers at least two, and let G be the pure braid group with n strands on a closed orientable surface of genus g. We describe any injective homomorphism from a finite index subgroup of G into G. As a consequence, we show that any finite index subgroup of G is co-Hopfian.

Study on twin groups with algorithm and properties.

We characterize co-Hopfian finitely generated torsion free nilpotent groups in terms of their Lie algebra automorphisms, and construct many examples of such groups.

Let B_n be the braid group on n > 3 strands. We prove that B_n modulo its center is co-Hopfian. We then show that any injective endomorphism of B_n is geometric in the sense that it is induced by a homeomorphism of a punctured disk. We further prove that any injection from B_n to B_n+1 is geometric. Additionally, we ob…

In all possible cases, we prove that local embeddings between two curve complexes whose complexities do not increase from domain to codomain are induced by surface homeomorphism. This is our first main result. From this we can deduce our second, a strong local co-Hopfian result for mapping class groups.

Let $N$ be at least 4. We prove that every injective homomorphism from the Torelli subgroup into $Out(F_N)$ differs from the inclusion by a conjugation in $Out(F_N)$. This applies more generally to the following subgroups: every finite-index subgroup of $Out(F_N)$ (recovering a theorem of Farb and Handel); every subgro…

Let S be a connected, compact and orientable surface of genus two having exactly one boundary component. We study automorphisms of the Torelli complex for S, and describe any isomorphism between finite index subgroups of the Torelli group for S. More generally, we study superinjective maps from the Torelli complex for …

Study of unrestricted virtual braid groups and their properties.

Tree-graded spaces are generalizations of R-trees. They appear as asymptotic cones of groups (when the cones have cut points). Since many questions about endomorphisms and automorphisms of groups, solving equations over groups, studying embeddings of a group into another group, etc. lead to actions of groups on the asy…

The paper classifies fixed subgroups of endomorphisms in free-abelian times surface groups.

Let K be the subgroup of the extended mapping class group, Mod(S), generated by Dehn twists about separating curves. Assuming that S is a closed, orientable surface of genus at least 4, we confirm a conjecture of Farb that Comm(K), Aut(K) and Mod(S) are all isomorphic. More generally, we show that any injection of a fi…

We show that for any positive integer $m\ge 1$, $m$-relator quotients of the modular group $M = PSL(2,\mathbb{Z})$ generically satisfy a very strong Mostow-type \emph{isomorphism rigidity}. We also prove that such quotients are generically "essentially incompressible". By this we mean that their "absolute $T$-invariant…

Let M be any closed, locally symmetric n-manifold (n>1) of nonpositive curvature. Assume that M has no locally Euclidean factors and no factors locally isometric to SL(3,R). Then for any closed Riemannian manifold N and any continuous map f:N -> M, we show there is a C^1 representative in the homotopy class of f with J…

Researchers compute BNSR-invariants for surface Houghton groups.

First the title could be also understood as ``3-manifolds related by non-zero degree maps" or "Degrees of maps between 3-manifolds" for some aspects in this survey talk. The topology of surfaces was completely understood at the end of 19th century, but maps between surfaces kept to be an active topic in the 20th centur…

We apply the method of Arzhantseva-Ol'shanskii to prove that for an exponentially generic (in the sense of Ol'shanskii) class of one-relator groups the isomorphism problem is solvable in at most exponential time. This is obtained as a corollary of our more general result that for any fixed integers $m>1, n>0$ there is …

Groups with Property (T) have fiber products with Property (T).

We prove recognition theorems for codimension one manifold factors of dimension $n \geq 4$. In particular, we formalize topographical methods and introduce three ribbons properties: the crinkled ribbons property, the twisted crinkled ribbons property, and the fuzzy ribbons property. We show that $X \times \mathbb{R}$ i…

The study shows that several properties are not profinite invariants.

We show that all finite-dimensional resolvable generalized manifolds with the piecewise disjoint arc-disk property are codimension one manifold factors. We then show how the piecewise disjoint arc-disk property and other general position properties that detect codimension one manifold factors are related. We also note …

The paper explores higher property T in lattices and its connections to geometric phenomena.

Asymptotic property C was introduced by Dranishnikov to study spaces with infinite asymptotic dimension. We show that asymptotic property C is preserved by infinite products. We also show that countable restricted direct products of countable groups with finite asymptotic dimension have asymptotic property C. Then we i…

Groups of importance in group theory have flexible stability properties.

This paper generalizes property (QT) to a broader class of groups.

Proves a vanishing property for symplectic manifold cohomology.

Survey on foliations and diffeomorphism groups.

We prove extension theorems for several geometric properties such as asymptotic property C (APC), finite decomposition complexity (FDC), strict finite decomposition complexity (sFDC) which are weakenings of Gromov's finite asymptotic dimension (FAD). The context of all theorems is a finitely generated group $G$ with a …

Study generalizes property elicitation to imprecise probabilities.

Introduces bounded scale measure and generalizes property A.

Study cohomological and metric properties of non-Kähler complex manifolds.

Model predicts price polarity of real estate properties using website information.

The paper explores the twisted Rokhlin property in mapping class groups of surfaces.

The balance property is crucial for insurance pricing, ensuring total actuarial price equals loss. Maximum likelihood GLMs fulfill it, but Lindholm-Wüthrich suggests three methods, with constrained GLM being superior.

We prove some general results about quasi-actions on trees and define Property (QFA), which is analogous to Serre's Property (FA), but in the coarse setting. This property is shown to hold for a class of groups, including $SL(n,\Z)$ for $n\geq 3$. We also give a way of thinking about Property (QFA) by breaking it down …

We present examples of metric spaces that are not Riemannian manifolds nor dimensionally homogeneous that satisfy the Tetrahedral Property. In spite of that, Euclidean cones over metric spaces with small diameter do not satisfy this property. We extend Sormani's Tetrahedral Property to a less restrictive property and p…

Study Finsler metric measure manifolds' concentration properties.

The paper examines curvature properties of twistor spaces.

Paper proves naturally reductive property is inaudible for certain manifolds.

Hybrid clustering combines partitional and hierarchical clustering for computational effectiveness and versatility in cluster shape. In such clustering, a dissimilarity measure plays a crucial role in the hierarchical merging. The dissimilarity measure has great impact on the final clustering, and data-independent prop…

3-manifold groups have a property that allows them to act on quasi-trees.

The average shadowing property is considered for set-valued dynamical systems, generated by parameterized IFS, which are uniformly contracting, or conjugacy, or products of such ones. We also prove that if a continuous surjective IFS F on a compact metric space X has the aver- age shadowing property, then every point x…

A property, or statistical functional, is said to be elicitable if it minimizes expected loss for some loss function. The study of which properties are elicitable sheds light on the capabilities and limitations of point estimation and empirical risk minimization. While recent work asks which properties are elicitable, …

Simplicial spheres have a weak Lefschetz property in characteristic 2.

In this article, we survey recent developments in defining the quasi-local mass in general relativity. We discuss various approaches and the properties and applications of the different definitions. Among the expected properties, we focus on the rigidity property: for a surface in the Minkowski spacetime, one expects t…

Groups with hyperbolic properties don't have strong Property (T).