arXiv research

Uniform undistortion in cyclic subgroups of certain groups.

New examples of subgroups in mapping class groups are found.

New groups from strand diagrams show polycyclic subgroups are virtually abelian and undistorted.

We consider linear groups which do not contain unipotent elements of infinite order, which includes all linear groups in positive characteristic, and show that this class of groups has good properties which resemble those held by groups of non positive curvature and which do not hold for arbitrary characteristic zero l…

The affine diffeomorphism group $\mathrm{Aff}(S,q)$ of a half-translation surface $(S,q)$ comprise the self-diffeomorphisms with constant differential away from the singularities. This group coincides with the stabiliser of the associated Teichmüller disc under the action of the mapping class group on Teichmüller space…

In this paper, we prove a limit set intersection theorem in relatively hyperbolic groups. Our approach is based on a study of dynamical quasiconvexity of relatively quasiconvex subgroups. Using dynamical quasiconvexity, many well-known results on limit sets of geometrically finite Kleinian groups are derived in general…

We prove a Morse Lemma for coarsely regular quasigeodesics in nonpositively curved symmetric spaces and euclidean buildings X. The main application is a simpler coarse geometric characterization of Morse subgroups of the isometry groups Isom(X) as undistorted subgroups which are coarsely uniformly regular. We show furt…

Explicitly generates right-angled Artin subgroups from mapping classes.

Combination theorem for PGF groups helps in constructing new examples and understanding their geometry.

We prove that finitely generated purely loxodromic subgroups of a right-angled Artin group $A(Γ)$ fulfill equivalent conditions that parallel characterizations of convex cocompactness in mapping class groups $\text{Mod}(S)$. In particular, such subgroups are quasiconvex in $A(Γ)$. In addition, we identify a milder cond…

In this note, we prove that a random extension of either the free group $F_N$ of rank $N\ge3$ or of the fundamental group of a closed, orientable surface $S_g$ of genus $g\ge2$ is a hyperbolic group. Here, a random extension is one corresponding to a subgroup of either Out$(F_N)$ or Mod$(S_g)$ generated by $k$ independ…

We prove that two countable locally finite-by-abelian groups G,H endowed with proper left-invariant metrics are coarsely equivalent if and only if their asymptotic dimensions coincide and the groups are either both finitely-generated or both are infinitely generated. On the other hand, we show that each countable group…

The paper studies automorphisms of RAAGs and RACGs, proving properties of their fixed subgroups.

New examples of distorted interval diffeomorphisms found.

Most distortion correction methods focus on simple forms of distortion, such as radial or linear distortions. These works undistort images either based on measurements in the presence of a calibration grid, or use multiple views to find point correspondences and predict distortion parameters. When possible distortions …

In this paper we prove that groups as in the title are convex cocompact in the mapping class group.

Connected components of Morse boundaries are studied in graph of groups.

Let X be a connected topological space admitting a universal cover. Let a be a degree one cohomology class on X. We define and study a two-cocycle on a group acting on X by homeomorphisms preserving the class a. We use this cocycle to investigate group actions on X. For example, we show that if an action preserves a Bo…

Let X be a path-connected topological space admitting a universal cover. Let Homeo(X,a) denote the group of homeomorphisms of X preserving degree one cohomology class a. We investigate the distortion in Homeo(X,a). Let g be an element of Homeo(X,a). We define a Nielsen-type equivalence relation on the space of g-invari…

We study geometric properties of stabilisers in the handlebody group. We find that stabilisers of meridians are undistorted, while stabilisers of primitive curves or annuli are exponentially distorted for large enough genus.

The paper studies quasimorphisms and distortion in homeomorphism groups of manifolds.

New findings on metric spaces with finite Nagata dimension.

Study large-scale geometry of graph braid groups via cubical structures.

The so-called great divergence in the income per capita is described in the Unified Growth Theory as the mind-boggling and unresolved mystery about the growth process. This mystery has now been solved: the great divergence never happened. It was created by the manipulation of data. Economic growth in various regions is…

Magnetic Brunn-Minkowski inequalities on Riemannian manifolds

Study finds a linear lower bound on conformal dimension for random hyperbolic groups.

Given a finite metric CW complex $X$ and an element $α\in π_n(X)$, what are the properties of a geometrically optimal representative of $α$? We study the optimal volume of $kα$ as a function of $k$. Asymptotically, this function, whose inverse, for reasons of tradition, we call the volume distortion, turns out to be an…

Let $F_n$ be the free group on $n$ generators and $Γ_g$ the surface group of genus $g$. We consider two particular generating sets: the set of all primitive elements in $F_n$ and the set of all simple loops in $Γ_g$. We give a complete characterization of distorted and undistorted elements in the corresponding $Aut$-in…

In this paper, we compute the subgroup distortion of all finitely generated subgroups of all finitely generated 3-manifold groups, and the subgroup distortion in this case can only be linear, quadratic, exponential and double exponential. It turns out that the subgroup distortion of a subgroup of a 3-manifold group is …

Regular subgroups of SL3(R) are identified and ruled out.

Study on braid group quotients by congruence subgroups.

The paper explores geometric finiteness in mapping class groups and constructs new examples of these subgroups.

Proposes a new method for finding non-redundant, standout subgroups in numeric datasets.

Proves Congruence Subgroup Property for two types of groups.

New method constructs non-quasiconvex subgroups in hyperbolic groups.

Characterizes knotted subgroups of Lie groups and provides examples.

Study subgroups of pro-$p$ PD^3 groups, finding specific conditions.

No hyperbolic group can have an infinite chain of free subgroups of fixed rank.

Robust subgroup discovery finds non-redundant, statistically significant subgroups.

Sparse GFA identifies disease factors in FTD subgroups.

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…

New lattices in higher dimensions have dense surface subgroups.

For a finitely generated group, there are two recent generalizations of the notion of a quasiconvex subgroup of a word-hyperbolic group, namely a stable subgroup and a Morse or strongly quasiconvex subgroup. Durham and Taylor defined stability and proved stability is equivalent to convex cocompactness in mapping class …

A new algorithm COVA-FC improves subgroup-fair clustering efficiency.

We associate cube complexes called completions to each subgroup of a right-angled Coxeter group (RACG). A completion characterizes many properties of the subgroup such as whether it is quasiconvex, normal, finite-index or torsion-free. We use completions to show that reflection subgroups are quasiconvex, as are one-end…

Study answers arithmeticity question for normal subgroup of lattices.

We show that, in an Artin-Tits group of spherical type, the intersection of two parabolic subgroups is a parabolic subgroup. Moreover, we show that the set of parabolic subgroups forms a lattice with respect to inclusion. This extends to all Artin-Tits groups of spherical type a result that was previously known for bra…

Characterizes groups arising as fixed subgroups of RAAG automorphisms.