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Damian Osajda

Publications and source records attributed to Damian Osajda.

At least 19 recordsLinked to original sources
arXiv2026

Drilling hyperbolic groups

Given a hyperbolic group $G$ and a maximal infinite cyclic subgroup $\langle g \rangle$, we define a {\it drilling of $G$ along $g$}, which is a relatively hyperbolic group pair $(\widehat{G}, P)$. This is inspired by the well-studied procedure of drilling a hyperbolic $3$--manifold along an embedded geodesic. We prove that, under suitable conditions, a hyperbolic group with $2$-sphere boundary admits a drilling where the resulting relatively hyperbolic group pair $(\widehat{G}, P)$ has relatively hyperbolic boundary $S^2$. This allows us to reduce the Cannon Conjecture (in the residually finite case) to a relative version, which is likely to be more tractable.

math.GT↗
arXiv2026

Fit systolic groups, exactly

A systolic complex/bridged graph is fit when its (metric) intervals are "not too large". We prove that uniformly locally finite fit systolic complexes have Yu's Property A. In particular, groups acting properly on such complexes have Property A, (equivalently) they are exact, and (equivalently) they are boundary amenable. As applications we show that groups from a class containing all large-type Artin groups, as well as all finitely presented graphical $C(3)$--$T(6)$ small cancellation groups, and finitely presented classical $C(6)$ small cancellation groups are exact. We also provide further examples. Our proof relies on a combinatorial criterion for Property~A due to Špakula and Wright.

math.GR↗
arXiv2025

Graphical small cancellation and hyperfiniteness of boundary actions

We study actions of (infinitely presented) graphical small cancellation groups on the Gromov boundaries of their coned-off Cayley graphs. We show that a class of graphical small cancellation groups, including (infinitely presented) classical small cancellation groups, admit hyperfinite boundary actions, more precisely, the orbit equivalence relation that they induce on the boundaries of the coned-off Cayley graphs is hyperfinite.

math.GR↗
arXiv2025

Locally elliptic actions, torsion groups, and nonpositively curved spaces

Extending and unifying a number of well-known conjectures and open questions, we conjecture that locally elliptic (that is, every element has a bounded orbit) actions by automorphisms of finitely generated groups on finite dimensional nonpositively curved complexes have global fixed points. In particular, finitely generated torsion groups cannot act without fixed points on such spaces. We prove these conjectures for a wide class of complexes, including all infinite families of Euclidean buildings, Helly complexes, some graphical small cancellation and systolic complexes, uniformly locally finite Gromov hyperbolic graphs. We present consequences of these result, e.g. concerning the automatic continuity. Our main tool are Helly graphs. On the way we prove several results concerning their automorphisms. We present and study a new notion of geodesic clique paths. Their local-to-global properties are crucial in our proof of ellipticity results.

math.GR↗
arXiv2024

Rigidity and classification results for large-type Artin groups

We compute the automorphism group of the intersection graph of many large-type Artin groups. This graph is an analogue of the curve graph of mapping class groups but in the context of Artin groups. As an application, we deduce a number of rigidity and classification results for these groups, including computation of outer automorphism groups, commensurability classification, quasi-isometric rigidity, measure equivalence rigidity, orbit equivalence rigidity, rigidity of lattice embedding, and rigidity of cross-product von Neumann algebra.

math.GR↗
arXiv2023

Helly groups

Helly graphs are graphs in which every family of pairwise intersecting balls has a non-empty intersection. This is a classical and widely studied class of graphs. In this article we focus on groups acting geometrically on Helly graphs -- Helly groups. We provide numerous examples of such groups: all (Gromov) hyperbolic, CAT(0) cubical, finitely presented graphical C(4)$-$T(4) small cancellation groups, and type-preserving uniform lattices in Euclidean buildings of type $C_n$ are Helly; free products of Helly groups with amalgamation over finite subgroups, graph products of Helly groups, some diagram products of Helly groups, some right-angled graphs of Helly groups, and quotients of Helly groups by finite normal subgroups are Helly. We show many properties of Helly groups: biautomaticity, existence of finite dimensional models for classifying spaces for proper actions, contractibility of asymptotic cones, existence of EZ-boundaries, satisfiability of the Farrell-Jones conjecture and of the coarse Baum-Connes conjecture. This leads to new results for some classical families of groups (e.g. for FC-type Artin groups) and to a unified approach to results obtained earlier.

math.GR↗
arXiv2022

Coxeter groups are biautomatic

We prove that Coxeter groups are biautomatic. From our construction of the biautomatic structure it follows that uniform lattices in isometry groups of buildings are biautomatic.

math.GR↗
arXiv2022

Helly groups, coarsely Helly groups, and relative hyperbolicity

A simplicial graph is said to be (coarsely) Helly if any collection of pairwise intersecting balls has non-empty (coarse) intersection. (Coarsely) Helly groups are groups acting geometrically on (coarsely) Helly graphs. Our main result is that finitely generated groups that are hyperbolic relative to (coarsely) Helly subgroups are themselves (coarsely) Helly. One important consequence is that various classical groups, including toral relatively hyperbolic groups, are equipped with a CAT(0)-like structure -- they act geometrically on spaces with convex geodesic bicombing. As a means of proving the main theorems we establish a result of independent interest concerning relatively hyperbolic groups: a `relatively hyperbolic' description of geodesics in a graph on which a relatively hyperbolic group acts geometrically. In the other direction, we show that for relatively hyperbolic (coarsely) Helly groups their parabolic subgroups are (coarsely) Helly as well. More generally, we show that `quasiconvex' subgroups of (coarsely) Helly groups are themselves (coarsely) Helly.

math.GR↗
arXiv2022

Torsion groups do not act on $2$-dimensional $\mathrm{CAT}(0)$ complexes

We show, under mild hypotheses, that if each element of a finitely generated group acting on a $2$-dimensional $\mathrm{CAT}(0)$ complex has a fixed point, then there is a global fixed point. In particular all actions of finitely generated torsion groups on such complexes have global fixed points. The proofs rely on Masur's theorem on periodic trajectories in rational billiards, and Ballmann-Brin's methods for finding closed geodesics in $2$-dimensional locally $\mathrm{CAT}(0)$ complexes. As another ingredient we prove that the image of an immersed loop in a graph of girth $2π$ with length not commensurable with $π$ has diameter $> π$. This is closely related to a theorem of Dehn on tiling rectangles by squares.

math.GR↗
arXiv2021

Normal subgroups of SimpHAtic groups

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 nonuniform 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.

math.GR↗
arXiv2021

Tits Alternative for groups acting properly on $2$-dimensional recurrent complexes (with an appendix written jointly with Jon McCammond)

We prove the Tits Alternative for groups acting on $2$-dimensional "recurrent" complexes with uniformly bounded cell stabilisers. This class of complexes includes, among others: $2$-dimensional Euclidean buildings, $2$-dimensional systolic complexes, $B(6)$-small cancellation complexes, and standard Cayley complexes for Artin groups of extra-large type. In the appendix written jointly with Jon McCammond we extend the result to a class of $2$-dimensional Artin groups containing all large-type Artin groups.

math.GR↗
arXiv2021

2-dimensional Coxeter groups are biautomatic

Let $W$ be a $2$-dimensional Coxeter group, that is, a one with $\frac{1}{m_{st}}+\frac{1}{m_{sr}}+\frac{1}{m_{tr}}\leq 1$ for all triples of distinct $s,t,r\in S$. We prove that $W$ is biautomatic. We do it by showing that a natural geodesic language is regular (for arbitrary $W$), and satisfies the fellow traveller property. As a consequence, by the work of Jacek Świątkowski, groups acting properly and cocompactly on buildings of type $W$ are also biautomatic. We also show that the fellow traveller property for the natural language fails for $W=\widetilde{A}_3$.

math.GR↗
arXiv2021

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 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 mean of proving the Helly property we introduce and use the notion of a (generalized) cell Helly complex.

math.GR↗
arXiv2020

Small cancellation labellings of some infinite graphs and applications

We construct small cancellation labellings for some infinite sequences of finite graphs of bounded degree. We use them to define infinite graphical small cancellation presentations of groups. This technique allows us to provide examples of groups with exotic properties: - We construct the first examples of finitely generated coarsely non-amenable groups (that is, groups without Guoliang Yu's Property A) that are coarsely embeddable into a Hilbert space. Moreover, our groups act properly on CAT(0) cubical complexes. - We construct the first examples of finitely generated groups, with expanders embedded isometrically into their Cayley graphs - in contrast,in the case of the Gromov monster expanders are not even coarsely embedded. We present further applications.

math.GR↗
arXiv2020

Quasi-Euclidean tilings over 2-dimensional Artin groups and their applications

We describe the structure of quasiflats in two-dimensio\-nal Artin groups. We rely on the notion of metric systolicity developed in our previous work. Using this weak form of non-positive curvature and analyzing in details the combinatorics of tilings of the plane we describe precisely the building blocks for quasiflats in all two-dimensional Artin groups -- atomic sectors. This allows us to provide useful quasi-isometry invariants for such groups -- completions of atomic sectors, stable lines, and the intersection pattern of certain abelian subgroups. These are described combinatorially, in terms of the structure of the graph defining an Artin group. As an important tool, we introduce an analogue of the curve complex in the context of two-dimensional Artin groups -- the intersection graph. We show quasi-isometric invariance of the intersection graph under natural assumptions. As immediate consequences we present a number of results concerning quasi-isometric rigidity for the subclass of CLTTF Artin groups. We give a necessary and sufficient condition for such groups to be strongly rigid (self quasi-isometries are close to automorphisms), we describe quasi-isometry groups, we indicate when quasi-isometries imply isomorphisms for such groups. In particular, there exist many strongly rigid large-type Artin groups. In contrast, none of the right-angled Artin groups are strongly rigid by a previous work of Bestvina, Kleiner and Sageev.

math.GR↗
arXiv2019

A combination theorem for combinatorially non-positively curved complexes of hyperbolic groups

We prove a combination theorem for hyperbolic groups, in the case of groups acting on complexes displaying combinatorial features reminiscent of non-positive curvature. Such complexes include for instance weakly systolic complexes and C'(1/6) small cancellation polygonal complexes. Our proof involves constructing a potential Gromov boundary for the resulting groups and analyzing the dynamics of the action on the boundary in order to use Bowditch's characterization of hyperbolicity. A key ingredient is the introduction of a combinatorial property that implies a weak form of non-positive curvature, and which holds for large classes of complexes. As an application, we study the hyperbolicity of groups obtained by small cancellation over a graph of hyperbolic groups.

math.GR↗