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Geometric Group Theory
Events

Submissions closed on 2025-02-14 11:59PM [Eastern Time (US & Canada)].

Organizers: Jone Lopez de Gamiz Zearra and Alice Kerr

Accepted Submissions:

Dehn Functions of Coabelian Subgroups — Pratit Goswami

The study of Dehn functions has developed into a major area of research in geometric group theory mainly because the growth types of these functions are quasi-isometry invariants of finitely presented groups. The Dehn function of a finitely presented group G is also connected to the complexity of solving the word problem in G namely, a finitely presented group has solvable word problem if and only if the Dehn function for a finite presentation is recursive. In this talk, we will discuss new methods for computing the precise Dehn functions of coabelian subgroups of direct products of groups, that is, subgroups which arise as kernels of homomorphisms from the direct product onto a free abelian group. This is joint work with Noel Brady and Rob Merrell.

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A weak Tits alternative for groups acting on buildings — Chris Karpinski

Buildings are highly symmetrical non-positively curved simplicial complexes introduced by Jacques Tits in the 1950s to study semisimple algebraic groups. Over the years, buildings have garnered interest among geometric group theorists due to their non-positively curved structure and close connections to Coxeter groups. We prove that groups acting properly and cocompactly on buildings satisfy an algebraic dichotomy, commonly encountered among groups with non-positive curvature features, known as the weak Tits alternative: either the group is virtually abelian or it contains a nonabelian free subgroup. This is joint work with Damian Osajda and Piotr Przytycki.

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Acylindrical actions on trees and applications to the outer automorphism group of Baumslag-Solitar groups — Bratati Som

An acylindrical action generalizes proper and cobounded actions on hyperbolic spaces. Non-elementary acylindrical actions provide acylindrically hyperbolic groups, which includes most mapping class groups of punctured surfaces, 3-manifold groups, and Out(Fn) for n>1. In this talk, we will explore how acylindricity of a group action on a tree can be preserved under quotients by certain subgroups, and discuss the existence of a largest acylindrical action for some groups acting on trees. In addition, we will show when Out(BS(p,q)) is acylindrically hyperbolic for non-solvable Baumslag-Solitar groups, despite BS(p,q) itself not being acylindrically hyperbolic, and explore further applications of these acylindricity results. This is a joint work with Daxun Wang.

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Anosov representations of cubulated hyperbolic groups — Theodore Weisman

An Anosov representation of a hyperbolic group Γ is a representation which quasi-isometrically embeds Γ into a semisimple Lie group - say, SL(d, R) - in a way which generalizes and imitates the dynamical behavior of a convex cocompact group acting on a hyperbolic metric space. It is unknown whether every linear hyperbolic group admits an Anosov representation. In this talk, after motivating the theory of Anosov representations from the perspective of geometric group theory, I will discuss joint work with Sami Douba, Balthazar Flechelles, and Feng Zhu which shows that every hyperbolic group acting geometrically on a CAT(0) cube complex admits a 1-Anosov representation into SL(d, R) for some d. The proof exploits the relationship between the combinatorial/CAT(0) geometry of right-angled Coxeter groups and the projective geometry of a convex domain in real projective space on which a Coxeter group acts by reflections.

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Asymptotic dimension of graphs of arcs and curves on infinite-type surfaces — Michael Kopreski

In analogy to the curve complex and its role in the geometry of mapping class groups of finite-type surfaces, a number of authors have defined graphs whose vertices are arcs or curves on a given infinite-type surface S, and on which the mapping class group Map(S) acts by isometries. We show that for a broad class of such graphs, including the grand arc graph, the omnipresent arc graph, and all others defined comparably to Masur-Minsky, the asymptotic dimension is infinite. In particular, if one could construct a graph in this class admitting a Švarc-Milnor-type action of Map(S), then Map(S) would have infinite asymptotic dimension.

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Automorphisms of the sphere complex of an infinite graph — Thomas Hill

For a locally finite, connected graph Γ, let Map(Γ) denote the group of proper homotopy equivalences of Γ up to proper homotopy. Excluding sporadic cases, we show Aut(S(MΓ))≅Map(Γ), where S(MΓ) is the sphere complex of the doubled handlebody MΓ associated to Γ. We also construct an exhaustion of S(MΓ) by finite strongly rigid sets when Γ has finite rank and finitely many rays, and an appropriate generalization otherwise. This is joint work with Michael Kopreski, Rebecca Rechkin, George Shaji, and Brian Udall.

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CAT(0) geometry of complex curve complements and families — Kejia Zhu

Motivated by the question of whether braid groups are CAT(0), we investigate the CAT(0) behavior of fundamental groups of plane curve complements and certain universal families. If C is the branch locus of a generic projection of a smooth, complete intersection surface to P2, we show that π1(P2∖C) is CAT(0). In the other direction, we prove that the fundamental group of the universal family associated with the singularities of type E6, E7, and E8 is not CAT(0). Other examples, both positive and negative, are discussed, with a special emphasis on rational 3-cuspidal curves. This is joint work with C. Bregman and A. Libgober.

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Centers of Artin Groups Defined on Cones — MurphyKate Montee

The Center Conjecture for Artin groups proposes that the center of any infinite type Artin group is trivial. This is known to hold for a wide class of Artin groups, but is not known in general. In this talk we will prove that the Center Conjecture passes to the Artin groups whose defining graphs are cones, if the conjecture holds for the Artin group defined on the set of the cone points. In particular, it holds for every Artin group whose defining graph has exactly one cone point. This is joint work with Kasia Jankiewicz.

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Characterizations of Stability via Morse Limit Sets — Jacob Garcia

An important example of Kleinian groups are the convex cocompact groups: every infinite order element of these groups is a loxodromic, and these groups are exactly the ones which admit Kleinian manifolds. A well known fact of convex cocompact groups is that they can be characterized exactly as the groups whose limit sets, on the visual boundary, are completely conical, or equivalently, completely horospherical. Convex cocompactness has been studied in the context of many non-hyperbolic spaces, such as mapping class groups, and has recently been generalized to the notion of subgroup stability. By using an analog of the visual boundary called the Morse boundary, a quasi-isometry invariant which "sees" hyperbolic directions for non-hyperbolic spaces, we show that subgroup stability is exactly classified by limit set conditions on the Morse Boundary which are analogous to the limit set conditions from the convex cocompact setting.

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Connected components in Morse boundaries of right-angled Coxeter groups — Annette Karrer

Every finitely generated group G has an associated topological space, called a Morse boundary, that captures the hyperbolic-like behavior of G at infinity. It was introduced by Cordes generalizing the contracting boundary invented by Charney--Sultan. In this talk, we study subgroups arising from connected components in Morse boundaries of right-angled Coxeter groups and of such that are quasi-isometric to right-angled Coxeter groups. This talk is based on two projects. One is joint work with Bobby Miraftab and Stefanie Zbinden. The other one is joint work in progress with Matthew Cordes and Kim Ruane.

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Distinguishing filling curve types via special metrics — Sayantika Mondal

In this talk, we look at filling curves on hyperbolic surfaces and consider its length infima in the moduli space of the surface as a type invariant. In particular, explore the relations between the length infimum of curves and their self-intersection number. For any given surface, we will construct infinite families of filling curves that cannot be distinguished by self-intersection number but via length infimum. I might also discuss some coarse bounds on the special metrics associated with these infimum lengths.

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Frattini subgroups of hyperbolic-like groups — Ekaterina Rybak

The Frattini subgroup Φ(G) of a group G is the intersection of all maximal subgroups of G; if G has no maximal subgroups, Φ(G)=G by definition. Frattini subgroups of groups with ``hyperbolic-like" geometry are often small in a suitable sense. Generalizing several known results, we prove that for any countable group G admitting a general type action on a hyperbolic space S, the induced action of the Frattini subgroup Φ(G) on S has bounded orbits, in particular, Φ(G) has infinite index in G. In contrast, we show that the Frattini subgroup of an infinite lacunary hyperbolic group can have finite index. The talk is based on a joint work with Gil Goffer and Denis Osin.

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Green metrics on hyperbolic groups and reparameterizations of the geodesic flow — Eduardo Reyes

Teichmüller space is a classical construction that, for a given closed hyperbolic surface, parameterizes the geometric actions of its fundamental group on the hyperbolic plane. I will talk about a generalization of this space, where for an arbitrary hyperbolic group we consider a space parameterizing its geometric actions on Gromov hyperbolic spaces, simultaneously encoding negatively curved Riemannian metrics, Anosov representations, random walks, geometric cubulations, etc. In particular, I will discuss how Green metrics (those encoding admissible random walks on the group) are dense in this space. As an application, for fundamental groups of negatively curved manifolds we produce a dictionary between this space of geometric actions and the space of reparameterizations of the geodesic flow. This is joint work with Stephen Cantrell and Dídac Martínez-Granado.

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Independence of Dehn, conjugator length, and annular Dehn functions of finitely presented groups — Conan Gillis

Brick and Corson introduced annular Dehn functions in 1998 to quantify the conjugacy problem for finitely generated groups and gave the fundamental relationships between it, the Dehn function, and the conjugator length function. I will discuss the key ideas behind these invariants, as well as joint work with T. Riley where we prove that these three invariants are independent—in general, no two of the three functions determine the other.

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Latent symmetry of graphs and stretch factors in Out(Fn) — Paige Hillen

Given an irreducible element of Out(Fn), there is a graph and an irreducible "train track map" on this graph, which induces the outer automorphism on the fundamental group. The stretch factor of an outer automorphism measures the asymptotic growth rate of words in Fn under applications of the automorphism, and appears as the leading eigenvalue of the transition matrix of such a train track representative. I'll present work showing a lower bound for the stretch factor in terms of the number of edges in the graph and the number of folds in the fold decomposition of the train track map. Moreover, in certain cases, a notion of the latent symmetry of a graph G gives a lower bound on the number of folds required for any irreducible train track map on G. I'll use this to classify all single fold train track maps.

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Left-invariant Riemannian distances on higher-rank Sol-type groups. — Daniel Levitin

Describing the coarse geometry of solvable groups is one of the major projects of geometric group theory. One solvable group whose geometry is well-understood is Sol, a rank-1 group foliated by two families of hyperbolic planes. More generally, Le Donne, Pallier, and Xie recently described the geodesics in Sol-type groups, which are the rank-1 solvable groups foliated by a pair of negatively-curved spaces. Leveraging this description, they show that all left-invariant Riemannian distances on a Sol-type group are roughly similar. In this talk, I will describe the coarse geometry of the broader class of higher-rank Sol-type groups, and discuss my generalization of Le Donne-Pallier-Xie's result to certain distances on these groups.

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Pattern preserving quasi-isometries in lamplighter groups — Beibei Liu

In this talk, I will explore the interplay between aspects of the geometry and algebra of three families of groups of the form B⋊Z, namely Lamplighter groups, solvable Baumslag-Solitar groups and lattices in SOL. In particular we examine what kind of maps are induced on B by quasi-isometries that coarsely permute cosets of the Z subgroup. By the results of Schwartz(1996) and Taback(2000) in the lattice in SOL and solvable Baumslag-Solitar cases respectively such quasi-isometries induce parallelogram preserving maps of B. We show that this is no longer true in the lamplighter case but the induced maps do share some features with parallelogram preserving maps. This is joint work with Dymarz, Macura and Morris-Wright.

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Random branched covers of 2-complexes — Jean-Francois Lafont

I'll introduce a random model for branched covers of 2-complexes. I'll explain why asymptotically almost surely, a random branched cover has Gromov hyperbolic fundamental group. This is joint work with Hyeran Cho (OSU) and Rachel Skipper (Univ. Utah).

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The RAAG Recognition Problem for Bestvina--Brady Groups — Yu-Chan Chang

Right-angled Artin groups (RAAGs) are an important class of objects of study in geometric group theory. It is interesting to know which groups are isomorphic to a RAAG. In this talk, we will explore how to recognize a Bestvina–Brady group as a RAAG. In particular, I will focus on Bestvina–Brady groups defined on 2-dimensional flag complexes. This is joint work with Lorenzo Ruffoni.

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