Kasia Jankiewicz

Title: Boundary rigidity for groups acting on products of trees

Abstract: gra The visual boundary is a well-defined compactification of a hyperbolic or CAT(0) space. For hyperbolic groups, the boundary is unique up to homeomorphism. However, Croke-Kleiner constructed examples of CAT(0) spaces with non-homeomorphic boundaries. I will discuss the question of the uniqueness of the boundary for groups acting geometrically on product of two trees. This is a wide family of groups including products of gree groups, as well as some simple groups. This is joint work with Annette Karrer, Kim Ruane, and Bakul Sathaye.

Carolyn Abbott

Title: Actions of Baumslag-Solitar groups on hyperbolic spaces

Abstract: A useful tool in understanding a group is to study its actions on hyperbolic metric spaces. Many groups admit lots of distinct, interesting actions on different hyperbolic spaces. In this talk, I will discuss how one can try to classify all actions of a given group on hyperbolic spaces, and why it is difficult to do so in general. I will then describe certain classes of groups for which this is possible, focusing on solvable Baumsag-Solitar groups. Surprisingly, this classification uses techniques from algebraic number theory. Finally, I will describe some work-in-progress which generalizes these methods to larger classes of groups. Parts of this talk are joint work with Alex Rasmussen, and parts are joint work with Sahana Balasubramanya and Alex Rasumusssen.

Jen Taback

Title: Conjugation curvature in solvable-Solitar groups

Abstract: Bar Natan, Duchin, and Kropholler introduced conjugation curvatures as a discrete Rici curvature for Cayley graphs of finitely generated groups. In joint work with Alden Walker, we show that the solvable Baumslag-Solitar groups BS(1,n) has sets of elements of positive, negative, and zero conjugation curvature, and that these sets have positive density in the group. To prove this, we use a lattice-based approach to produce a geodesic representative for every element of the group, from which we derive a word-length formula, all with respect to the standard generating set for BS(1,n). A subset of these geodesic representatives forms a regular language, and in subsequent we use this language to give a new proof of the growth rate of BS(1,n).

Volodymyr Nekrashevych

Title: Hyperbolic groupoids

Abstract: We will discuss a natural notion of hyperbolicity for groupoids (and semigroups), which generalizes actions of hyperbolic groups on their boundaries, groupoids associated with contracting self-similar groups, and groupoids associated with stable and unstable foliations of Anosov diffeomorphisms. We will also discuss some intriguing open questions related to it.

Daniel Studenmund

Title: Topological models for abstract commensurator groups

Abstract: As the automorphism group of a group G encodes its symmetries, the abstract commensurator group of G encodes its "hidden symmetries". Actions of commensurator groups on spaces, such as the action of the commensurator of a mapping class group on an appropriate curve complex, often arise in connection to rigidity results. On the other hand, commensurators of non-rigid groups, such as surface groups and free groups, tend not to have well-understood topological models. Odden proved that the abstract commensurator of a higher-genus closed surface group is isomorphic to a group of mapping classes of the universal hyperbolic solenoid, the inverse limit of the system of finite covers of the genus 2 surface. This talk will review these results, and discuss work in progress realizing commensurators of certain nonpositively curved groups as self-homotopy equivalences of solenoids. Joint with Edgar A. Bering IV.

Sean Cleary

Title: Exploring for typical phenomena in Thompson's groups

Abstract: Thompson's groups F, T, and V have a wide range of types of elements, isomorphism classes of subgroups, and measures of cancellation. We describe some ways of understanding what is "typical behavior" in these groups in the sense of which phenomena have non-zero limits for density in measures of increasing size of instances.

Tarik Agouab

Title: Thermodynamic metrics on outer space

Abstract: The Weil-Petersson metric on Teichmuller space can be obtained by studying the dynamics of geodesic lengths on a surface as one infinitesimally deforms the metric. In this talk, we use this dynamical characterization to explore an analog of the Weil-Petersson metric on the outer space. We show that, like in the surface case, this metric is always incomplete and we characterize the completion of the top dimensional cells corresponding to roses. We also show that there is a global fixed point for the Out(F_n) action. This represents joint work with Matt Clay and Yo'av Rieck.

Olga Kharlompovich

Title: Fraisse limits of limit groups

Abstract: We modify the notion of a Fra\"{i}ss\'{e} class and show that various interesting classes of groups, notably the class of nonabelian limit groups and the class of finitely generated elementary free groups, admit Fra\"{i}ss\'{e} limits. We rediscover Lyndon's $\Z[t]$-exponential completions of countable torsion-free CSA groups, as Fra\"{i}ss\'{e} limits with respect to extensions of centralizers.

We will also discuss countable elementary free groups. These are joint results with A. Miasnikov, R. Sklinos, C. Natoli.

Thomas Koberda

Title: The virtual critical regularity of mapping class group actions on the circle

Abstract: The classical work of Nielsen and Thurston-Handel shows that the mapping class group of a surface with a marked point acts faithfully by homeomorphisms on the circle. In this talk, I will discuss (and resolve) the problem of finding the sharpest regularity with which a finite index subgroup of the mapping class group can act faithfully on the circle. This talk represents joint work with S. Kim and C. Rivas.

Nicolas Monod

Title: Asymptotic isoperimetry on Cayley graphs and unitarizable representations

Abstract: Joint work with Gersimov-Gruber-Thom.

Folner's theorem states that a finitely generated group G is amenable if and only if a basi isoperimetric inequality is satisfied by the Cayley graph associated to some (or any) finite generating set S of G. In particular, when G is non-amenable, every S gives us a positive isoperimetric constant. We define an asymptotic isopermetric invariant by examining how this constant varies when S increases. We prove that this asymptotic invariant is related to understand when a representation of G is unitarizable. This is motivated by Dixmier's problem, open since 1950, namely: is amenability the same as unitarizability?

Tullia Dymarz

Title: Quasi-isometric rigidity and non-rigidity of Gersten groups

Abstract: Recently there has been a lot of work on the large scale geometry and quasi-isometric rigidity of various hyperbolic or CAT(0) groups that split over cyclic subgroups. In contrast, we look at groups built out of solvable Baumslag-Solitar groups glued along their cyclic subgroups. The simplest of these examples is the so-called Baumslag-Gersten group, a one relator group that has been well studied by combinatorial group theorists.

Ruth Charney

Title: The topology of Morse boundaries

Abstract: The Morse boundary was introduced to encode hyperbolic-like behavior in non-hyperbolic geodesic metric spaces. it is a quasi-isometry invariant, hence well-defined for any finitely generated group. While many properties of Morse boundaries have been established in recent years, there are few examples of non-hyperbolic groups for which there is a complete topology description of this boundary. I will discuss joint work with M. Cordes and A. Sisto giving precise descriptions of the Morse boundaries of right-angled Artin groups, fundamental groups of non-geometric graph, and finite-volume cusped hyperbolic 3-manifolds.

Xiangdong Xie

Title: Groups whose left invariant metrics are all roughly similar

Abstract: wwo metrics $d_1$, $d_2$ on a set $X$ are similar if there is a constant $\lambda>0$ such that $d_2(x,y)=\lambda d_1(x,y)$ for all $x, y\in X$. Two metrics $d_1$, $d_2$ on a set $X$ are roughly similar if there are constants $\lambda>0, C\ge 0$ such that $\lambda d_1(x,y)-C\le d_2(x,y)\le \lambda d_1(x,y)+C$ for all $x, y\in X$. In Euclidean spaces (of dimension at least two) there are lots of norms so that the associated metrics are not roughly similar. We exhibit a class of solvable Lie groups (including Heintze groups and SOL type groups) where every two left invariant Riemmanian metrics are roughly similar. This is a new phenomenon. It is open whether any discrete groups (other than virtually cyclic groups) have this property. This is joint work with Enrico Le Donne and Gabriel Pallier.

Matt Zaremsky

Title: Geometric embeddings into simple groups

Abstract: In this talk I will discuss a generalization of Thompson's groups, called "twisted Brin-Thompson groups," which were introduced in 2020 by Jim Belk and myself and have a variety of interesting applications. First, we prove that every finitely generated group embeds (quasi-)isometrically as a subgroup of a finitely generated simple group, strengthening a result of Bridson. Second, we construct the second known infinite family of simple groups with distinct finiteness properties, following examples due to Rachel Skipper, Stefan Witzel, and myself using Roever-Nekrashevych groups. Lastly, we find examples of simple groups of type F_\infty containing every virtually special group.

Arman Darbinyan

Title: Embeddings into left-orderable simple groups

Abstract: A recent advancement in the theory of left-orderable groups is the discovery of finitely generated left-orderable simple groups by Hyde and Lodha. We will discuss a construction that extends this result by showing that every countable left-orderable group is a subgroup of such a group. We will also discuss some of the algebraic, geometric, and computability properties that this construction bears. In particular, the embeddings that we construct preserve computability properties of left-orders on initial group, are quasi-isometric, Frattini, etc. We will also discuss a version of Boone-Higman-Thompson criterion on the decidability of the word problem in groups in case the group is left-orderable with recursively enumerable positive cone. The flexibility of the embedding method allows us to go beyond the class of left-orderable groups as well. Based on a joint work with Markus Steenbock.

Robert Young

Title: Metric embeddings of the Heisenberg group

Abstract: The Heisenberg group is the simplest example of a noncommutative nilpotent Lie group, and that noncommutativity leads to surprisingly rich geometry. In this talk, we will explore the geometry of the Heisenberg group: how the word metric limits to a sub-Riemannian metric, how Pansu and Semmes used a version of Rademacher's differentiation theorem to show that there is no bilipschitz embedding from the Heisenberg groups into Euclidean space, and how the noncommutativity of the Heisenberg group makes it difficult to embed into Banach spaces.

Turbo Ho

Title: Languages of geodesics and rational growths of groups

Abstract: Given a group G with a finite generating set S, a representative system R is a set of words in S such that every element of G is represented by a unique element of R. Some representative systemes are nicer than others. For instance, the Mal'cev coordinate of a polycyclic group is a particularly nice representative system. This "niceness" can be formalized in several different ways, and when the representative system consists of geodesic words, one way to do this is via the rational growth of groups.

The rationality of hyperbolic groups, virtually abelian groups, and certain nilpotent groups have been studied by various people. On the other hand, the rational growths of some solvable groups are also established, and in many cases, this is done by finding a nice language of geodesics. We will discuss some past and recent results, and how the languages of geodesics may be useful in other work.

Francis Wagner

Title: Torsion Subgroups of Groups with Quadratic Dehn Function

Abstract: The Dehn function of a finitely presented group, first introduced by Gromov, is a useful invariant that is closely related to the solvability of the group’s word problem. It is well-known that a finitely presented group is word hyperbolic if and only if it has sub-quadratic (and thus linear) Dehn function. A result of Ghys and de la Harpe states that no word hyperbolic group can have a (finitely generated) infinite torsion subgroup. We show that this property does not carry over to any class of groups of larger Dehn function, producing the first examples of groups with quadratic Dehn function that contain a finitely generated infinite torsion subgroup. In particular, for every m>1 and every sufficiently large n (and either odd or divisible by 2^9), we produce a quasi-isometric embedding of the infinite free Burnside group B(m,n) into a finitely presented group with quadratic Dehn function.

Jone Lopez de Gamiz

Title: Subgroups of direct products of RAAGs

Abstract: Subgroups of direct products of (limit groups over) free groups have been extensively studied due to their importance in algebraic geometry over free groups, and under some finiteness properties, they have a simple structure. The class of right-angled Artin groups (RAAGs) extends the class of finitely generated free groups, so it is natural to try to understand subgroups of direct products of RAAGs. The goal of the talk is to motivate and to state the known results for subgroups of direct products of (limit groups over) free groups and to explain some generalizations in the class of RAAGs.

Yassine Guerch

Title: Rigidity of the automorphism group of a universal Coxeter group

Abstract: A universal Coxeter group of rank n, denoted by $W_n$, is the free product of n copies of a cyclic group of order 2. In this talk I will present some rigidity results for the outer automorphism group $Out(W_n)$ of a universal Coxeter group of rank n. Namely, I will sketch a proof that every automorphism of the outer automorphism group of a universal Coxeter group of rank at least 5 is induced by a global conjugation and, more generally, every isomorphism between two finite index subgroups of $Out(W_n)$ with n at least 5 is induced by a global conjugation. These algebraic rigidity results rely on the study of the action of $Out(W_n)$ on the Guirardel-Levitt spine of Outer space of $W_n$, whose symmetries are exactly described by elements of $Out(W_n)$.

Thomas Ng

Title: Efficient free subgroups in group extensions

Abstract: We say that a subgroup is efficiently generated when a copy can be generated by words whose word length is uniformly bounded over all generating sets. Efficient generation is a central tool in determining which groups have uniform exponential growth and is particularly natural is the setting of non-positive curvature. For example, Jankiewicz uses these ideas to exhibit a family of cubulated groups which cannot act properly on CAT(0) cube complexes of fixed dimension. In this talk, I will discuss joint work with Robert Kropholler and Rylee Lyman that describes when efficient generation of free subgroups is inherited by group extensions. One consequence is that Jankiewicz’s approach does not exhibit fibered 3-manifold groups with arbitrarily high cubical dimension. Our methods can further be used to show uniform exponential growth for automorphism groups of one-ended (total relatively) hyperbolic groups.

Casey Donoven

Title: Thompson's Group V is 3/2-Generated

Abstract: A group G is 3/2-generated if every nontrivial element of G is contained in a two-element generating set of G. Recent work by Burness, Guralnick, and Harper confirmed the conjecture that a finite group G is 3/2-generated if and only if every proper quotient of G is cyclic. This is unfortunately not the case for infinite group as there are many infinite simple groups that are not 3/2-generated. In this talk, I will present joint work with Scott Harper in which we show Thompson's Group V (and many more related groups) is 3/2-generated with a highly constructive proof.

Sam Hughes

Title: Groups quasi-isometric to right-angled Artin groups

Abstract: We will introduce quasi-isometric rigidity for groups and survey some results regarding right-angled Artin groups (RAAGs). We will then show how quasi-isometric rigidity fails for many RAAGs with centre of rank at least 2.

Jordan Bounds

Title: On Groups of Exponent 8

Abstract: Burnside's famous question asks whether or not every finitely generated group with finite exponent must be finite. While this need not be the case in general, there are several interesting cases where the answer is still unknown. In this talk we will discuss the unique case of groups of exponent 8 and some progress that has been made in this regard.

Jeroen Schillewaert

Title: Fixed points for group actions on 2-dimensional affine buildings

Abstract: We prove a local-to-global result for fixed points of groups acting on 2-dimensional affine buildings (assumed crystallographic when non-discrete). In the discrete case, our theorem establishes the corresponding special cases of a conjecture by Marquis. (joint work with Koen Struyve and Anne Thomas)

Dounnu Sasaki

Title: Currents on cusped hyperbolic surfaces and denseness property

Abstract: The space GC(S) of geodesic currents on a hyperbolic surface S can be considered as a completion of the set of weighted closed geodesics on S when S is compact, since the set of rational geodesic currents on S, which correspond to weighted closed geodesics, is a dense subset of GC(S). We proved that even when S is a cusped hyperbolic surface with finite area, GC(S) has the denseness property of rational geodesic currents, which correspond not only to weighted closed geodesics on S but also to weighted geodesics connecting two cusps. In addition, we proved that the space of subset currents on a cusped hyperbolic surface, which is a generalization of geodesic currents, also has the denseness property of rational subset currents.

Shi Wang

Title: Kleinian groups of small critical exponent

Abstract: Given a finitely generated, non-elementary discrete subgroup G<Isom(H^n), the G-orbits on H^n grows exponentially, and the exponential growth rate is defined to be the critical exponent of G. We show that, if the critical exponent is small enough, then G is convex cocompact, or equivalently the orbit map is a quasi-isometric embedding. This is joint work with Beibei Liu.

Zoran Sunic

Title: Symbolic dynamics of regular branch groups

Abstract: We prove that the topological closure of every self-replicating, regular branch group branches over a level stabilizer. In addition, we provide a description of the rigid kernel (the kernel of the completion with respect to rigid level stabilizers). We construct variations of the Hanoi Towers group on three pegs that provide the first examples of finitely generated groups with infinite rigid kernels.

The proof relies on symbolic dynamics on rooted regular trees. Namely, we prove that the topological closure of every self-replicating, regular branch group is a finitely constrained group, that is, a tree-shift of finite type. More generally, the completion with respect to rigid level stabilizers of every regular branch group is a tree-shift of finite type, and the topological closure is a sofic tree-shift.

(Joint work with Alejandra Garrido)

Srivatsav Kunnawalkam Elayavalli

Title: Proper proximality of groups acting on trees

Abstract: We show that countable groups acting by isometries on trees with mild malnormality conditions are properly proximal. This is a notion introduced by Boutonnet-Ioana-Peterson in the context of rigidity phenomena associated to the group von Neumann algebras. We also provide a complete classification of proper proximality for graph products of groups. This is joint work with Changying Ding.

Liam Stott

Title: A class of increasing homeomorphism groups naturally isomorphic to diagram groups

Abstract: Deep connections between groups of increasing homeomorphisms and diagram groups are suggested when one considers properties and examples of groups from each class. For example, work of Brin shows that Thompson’s group F embeds into a group of increasing piecewise linear homeomorphisms under a remarkably weak condition, while work of Guba and Sapir shows that F embeds into a diagram group under a similarly weak condition.

There have been limited results concerning direct connections; Guba and Sapir develop a procedure to find natural representations of any diagram group as a group of increasing homeomorphisms, though they are not always faithful. Thus, it is reasonable to ask: for which groups of increasing homeomorphisms does there exist a natural faithful representation as a diagram group? In this talk I will introduce a class of such groups, which includes groups generated by geometrically fast sets of one orbital homeomorphisms.

Funda Gultepe

Title: A universal Cannon-Thurston map and the surviving curve complex

Abstract: Using the Birman exact sequence for pure mapping class groups, we construct a universal Cannon-Thurston map onto the boundary of a curve complex for a surface with punctures we call surviving curve complex. Along the way we prove hyperbolicity of this complex and identify its boundary as a space of laminations. As a corollary we obtain a universal Cannon-Thurston map to the boundary of the ordinary curve complex. Joint work with Chris Leininger and Witsarut Pho-on.

Sushil Bhunia

Title: Twisted conjugacy in (big) mapping class groups

Abstract: Let φ be an automorphism of a group G. Two elements x and y of G are said to be φ-twisted conjugate if gx =yφ(g) for some g in G. This is an equivalence relation on G, and the equivalence classes are called the φ-twisted conjugacy classes or the Reidemeister classes of φ. If φ = Id then the φ-twisted conjugacy classes are the usual conjugacy classes. A group G is said to have the R_{∞} -property if the number of its φ-twisted conjugacy classes is infinite for every automorphism φ of G. In this talk I will describe when a big mapping class group (i.e., mapping class group of an infinite type surface) has the R_{∞} -property. This is a work in progress with Swathi Krishna.

Gangotryi Sorcar

Title: Conjugacy and reversibility in the group of plane homeomorphisms

Abstract: This talk is on a recent work with Sushil Bhunia, where we study self-homeomorphisms of the plane, and investigate topological or geometric conditions for them to be conjugate to each other. In particular, we are interested in understanding when they are conjugate to their own inverses (an element conjugate to its inverse is called reversible). Reversibility has been well studied in linear groups and isometry groups of certain symmetric spaces, but not much is known for homeomorphism groups of topological spaces.

Arye Juhasz

Title: A solution of the word problem in locally reducible Artin groups

Abstract: Let A be an Artin group with standard generating set X . A is called locally reducible if every three generated spherical standard parabolic subgroup is isomorphic to < a ,b ,c | R,S,T > where R=aba^-1b^-1,S=aca^-1c^-1 and T is an Artin relator on b and c of length at least 6.This class of Artin groups was introduced by Ruth Charney in connection with the Tits' Conjecture. In this talk we show that locally reducible Artin groups have polynomial isoperimetric function ( n^6 ) hence have solvable word problem . We introduce three notions for the proof concerning van Kampen diagrams ( hence presentations and groups ).1. The small cancellation condition A( 3) which is a kind of mixture of the classical small cancellation conditions. 2.The notion of residually small cancellation diagrams and show that locally reducible Artin groups are residually A( 3 ). 3. The notion of small cancellation- like diagrams. These are diagrams which are not small cancellation in any sense, yet they carry the basic properties of small cancellation diagrams , in particular they have polynomial Dehn functions. Finally ,we show that residually small cancellation diagrams are small cancellation-like.

Ignat Soroko

Title: Groups of type FP: their quasi-isometry classes and homological Dehn functions

Abstract: There are only countably many isomorphism classes of finitely presented groups. Considering a homological analog of finite presentability, we get the class of groups $FP_2$. Ian Leary proved that there are uncountably many isomorphism classes of groups of type $FP_2$. R.Kropholler, Leary and I proved that there are uncountably many classes of groups of type $FP_2$ even up to quasi-isometries. Since `almost all' of these groups are infinitely presented, the usual Dehn function makes no sense for them, but the homological Dehn function is well-defined. In a joint paper with N.Brady, R.Kropholler, we show that for any even integer $k\ge4$ there exist uncountably many quasi-isometry classes of groups of type $FP_2$ with a homological Dehn function $n^k$. In particular there exists an $FP_2$ group with the quartic homological Dehn function and the unsolvable word problem.

Bakul Sathaye

Title: A new large scale invariant of pairs Coarse knotting obstructions to Riemannian smoothings

Abstract: I will discuss a large scale quasi-isometry invariant of pairs of spaces (X,Y) called the "fundamental group of the Y-end" and then show it can be used distinguish certain smooth CAT(0) manifolds from Riemannian manifolds of nonpositive sectional curvature. This is joint work with Jean-Francois Lafont.

Jacob Russell

Title: Morse Local-to-Global Groups

Abstract: Quasi-geodesics in hyperbolic groups enjoy both a local-to-global property (every local quasi-geodesic is a global quasi-geodesic) and the Morse property (every quasi-geodesic is uniformly close to a geodesic). These properties not only play a central role in the theory of hyperbolic group, but each can be used to characterize hyperbolicity. Many groups beyond hyperbolic groups contain quasi-geodesics that satisfy the Morse property. While these Morse quasi-geodesics exhibit many features of hyperbolicity, they noticibly fail to have the local-to-global property present in hyperbolic groups.

We introduce the class of "Morse local-to-global" groups where the Morse geodesic do satisfy the local-to-global property. We show this class is much larger than hyperbolic groups (e.g. it contains all CAT(0), hierarchically hyperbolic, and 3-manifolds groups), but is still sufficiently power to secure result that would be untenable without a local-to-global property. We will survey the landscape of Morse local-to-global groups and discuss their applications to subgroup combinations and subgroup growth. Joint work with Davide Spriano, Hung C. Tran, Matt Cordes, and Abdul Zalloum.

Anschel Schaffer-Cohen

Title: Graphs of curves quasi-isometric to big mapping class groups

Abstract: This talk will present a graph of curves, the translatable curve graph, which is quasi-isometric to the mapping class groups of certain infinite-type surfaces. The obvious next step is to study the coarse geometry of this graph; collaboration in this area is encouraged.

Josiah Oh

Title: QI rigidity of lattice products

Abstract: Schwartz proved quasi-isometric rigidity for non-uniform lattices in rank one Lie groups. Frigerio–Lafont–Sisto later proved QI rigidity for products $\pi_1(M) \times \mathbb{Z}^d$ where $M$ is a complete, non-compact, finite-volume real hyperbolic manifold of dimension at least 3. This talk will cover QI rigidity for products $\Lambda \times L$, where $\Lambda$ is a non-uniform lattice in a rank one Lie group and $L$ is a lattice in a simply connected nilpotent Lie group. Specifically, any finitely generated group quasi-isometric to such a product is, up to some finite noise, an extension of a non-uniform rank one lattice by a nilpotent lattice. Under some extra hypotheses, this extension is (virtually) nilcentral, a notion we introduce as a generalization of central extensions.

Michal Ferov

Title: Conjugacy depth function for lamplighter groups

Abstract: Given a finitely generated group, its conjugacy depth function measures how deep within the lattice of normal subgroups of finite index one needs to go in order to be able to decide whether two elements are conjugate. In this talk I will sketch a proof lamplighter groups have exponential conjugacy depth function. (joint work with Mark Pengitore)

Lorenzo Ruffoni

Title: Graphical splittings of Artin kernels

Abstract: A main feature of the theory of right-angled Artin groups (RAAGs) consists in the fact that the algebraic properties of the group can be described in terms of the combinatorial properties of the underlying graph. We investigate how this can be exploited in the study of Artin kernels, i.e. subgroups of RAAGs obtained as kernels of integral characters. In the case of chordal graphs we obtain a sharp dichotomy for Artin kernels. For block graphs we obtain an explicit rank formula, and discuss some applications to the study of fibrations and BNS invariants of RAAGs. (Joint work with M. Barquinero and K. Ye).

Waltraud Lederle

Title: The minimal degree of a Cayley--Abels graph for Aut(T)

Abstract: We show that the automorphism group of a d-regular tree can not act vertex-transitively with compact, open vertex stabilizers on a connected graph of degree smaller than d (unless d = 1). In most cases, this is just a consequence of simplicity of the alternating group on d-1 letters. This gives the special case d=5, which is surprisingly tricky.

Burns Healy

Title: Group boundaries under semidirect products with the integers

Abstract: Given a group G that admits a Z-structure, we demonstrate a way to explicitly build a Z-structure for any group of the form G semidirect product with the integers. This procedure preserves many desired structures of the space and gives an exact description of the Z-set. As applications, we show one natural way to define group boundaries for all finitely generated nilpotent groups and all 3-manifold groups. Under mild hypotheses, these results extend to EZ-structures.

Annette Karrer

Title: Contracting boundaries of right-angled Coxeter groups

Abstract: Every complete CAT(0) space has a topological space associated to it, called the contracting or Morse boundary. This boundary indicates how similar the CAT(0) space is to a hyperbolic space. Charney--Sultan proved this boundary is a quasi-isometry invariant, i.e. it can be defined for CAT(0) groups. Interesting examples arise among right-angled Artin groups (RAAGs) and right-angled Coxeter groups (RACGs). Each such group is defined by a finite, simplicial graph, and acts geometrically on an associated CAT(0) cube complex. Despite these similarities, the contracting boundaries of RAAGs and RACGs behave differently. Charney--Cordes--Sisto showed that the contracting boundary of every RAAG is totally disconnected. Moreover, they showed that it is empty, a Cantor space, an omega-Cantor space, or consists of two points. In contrast to RAAGs, many diffrent topological spaces arise as Morse boundaries of RACGs and it is often difficult to determine whether a RACG has totally disconnected contracting boundary or not. In this talk, I will speak about the main result of my Ph.D. thesis that leads to a new class of RACGs with totally disconnected contracting boundaries. This class is obtained by generalizing an example of Charney--Sultan. Moreover, I will explain the relation to a joint project with Marius Graeber, Nir Lazarovich, and Emily Stark about surprising circles in contracting boundaries of RACGs.

MurphyKate Montee

Title: Cubulation and Property (T) in Random Groups

Abstract: The study of random groups is one way to answer the question, “What does a ‘typical’ group look like?” In the Gromov model of random groups, we choose a group presentation by picking cyclically reduced relators of a given length uniformly at random, where the number of relators is controlled by a quantity called the density. For some properties there is a sharp threshold at which the probability of satisfying the property switches from 1 to 0. Currently, sharp thresholds for Property (T) and cubulation are unknown, but we have some bounds. In this talk I’ll discuss recent work that suggests ways to sharpen the bounds for both properties, and show that with overwhelming probability for densities d < 3/14 random groups act non-trivially on a CAT(0) cube complex.

Mitul Islam

Title: Rank One Phenomena in Convex Projective Geometry

Abstract: A Hilbert geometry is a properly convex domain (i.e. subset of the real projective space that can be realized as a bounded convex domain in an affine chart) equipped with a projective cross-ratio distance function called the Hilbert metric. This is a generalization of the Beltrami-Klein model of the real hyperbolic spaces. However, a generic Hilbert geometry does not conform with any of the common notions of non-positive curvature like CAT(0). In this talk, we will introduce a notion of ``rank one" in Hilbert geometry and develop analogies with rank one CAT(0) geometry. Our main result is that rank one groups (in Hilbert geometry) are either acylindrically hyperbolic or contain a finite index cyclic subgroup. This leads to applications like counting conjugacy classes in rank one groups and infinite-dimensionality of the space of quasi-morphisms.

David Hume

Title: Kolmogorov-Barzdin estimates for three-dimensional real hyperbolic space

Abstract: In a remarkable paper in the 60's, Kolmogorov and Barzdin prove that any n-vertex graph may be "wired" into a ball of radius ~n^{1/2} in three-dimensional Euclidean space, and that the quantity of "wire" required is at most n^{3/2}. They then prove that these bounds are optimal for random 6-regular bipartite graphs. Implicit in their argument is the fact that these random graphs are expanders.

Estimating this "wiring complexity" of a network has modern applications. To perform sophisticated modern machine learning algorithms it is typically necessary to represent data in Euclidean space and use the linear structure (projections, hyperplanes, tensors etc.) within the algorithm. Very recently, it has been shown that some of these algorithms can also function in other spaces - most notably real hyperbolic space, and that for certain types of "tree-like" data, hyperbolic representations yield better results. For other types of data, Euclidean or spherical representations are the most useful.

I will prove that any n-vertex graph may be "wired" into a ball of radius ~log(n) in three-dimensional real hyperbolic space, that the quantity of "wire" required is at most n^2, and that this is optimal for expanders. This is joint work with Ben Barratt.