Accepted (20):
A Combinatorial Characterization of Sol 3-Manifolds — Leslie Mavrakis <l.mavrakis@utah.edu>
A family F of compact n-manifolds is locally combinatorially defined (LCD) if there is a finite collection of triangulated n-balls (called models) such that the set F is exactly the set of compact n-manifolds that have a triangulation which locally looks like one of these models. In previous work with Daryl Cooper and Priyam Patel, we show that being LCD is equivalent to the existence of a compact branched n-manifold W, such that F is precisely those manifolds that immerse into W. In this way, W can be thought of as a universal branched manifold for F. In this talk, I will explain why the set of Sol 3-manifolds is LCD by constructing a universal branched manifold for the family. The construction is based on regular languages that detect Anosov monodromies of torus bundles and the gluing maps of Sol semi-bundles.
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A language-theoretic characterization of f.g. subgroups of Thompson V — Davide Perego <dperego9@gmail.com>
The intersection of formal language theory and group theory provides a fascinating lens for studying algebraic structures, most notably through the word problem. This connection has allowed mathematicians to classify groups based on the Chomsky hierarchy. While the landmark Muller-Schupp theorem completely characterized groups with context-free word problems, progressing beyond this boundary has remained a major challenge. In this talk, we shift toward a more combinatorial and geometric approach. We will introduce a new framework that yields a characterization of the f.g. subgroups of Thompson V. Notably, this group is central to a well-known 2008 conjecture regarding groups with co-context-free word problems. Joint works with Corentin Bodart, Daniele D'Angeli, Francesco Matucci and Emanuele Rodaro.
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A linearity criterion for automorphism groups of hyperbolic groups — Mark Pengitore <mpengito@gmail.com>
This talk will introduce various growth functions associated to a finitely generated group which measure the difficulty of separating an element from the identity using epimorphisms to a fixed family of nonabelian finite simple groups with characteristic kernels as a function of the word length. As an application of these functions, we provide a characterization of when the automorphism group of a hyperbolic group is linear.
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Acyclicity of homeomorphism groups of stable Stone spaces — Michael Kopreski <michaelkopreski@gmail.com>
A stable second-countable Stone space is a closed subspace of the Cantor space with nice local structure. Examples arise as the end spaces of stable infinite-type surfaces and graphs. We classify the acyclicity of the homeomorphism groups of such spaces and describe progress toward computing their homology when not acyclic. These results are joint work with Mladen Bestvina and Rachel Skipper.
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Algebraic fibrations, hyperbolic Coxeter groups and a hidden icosahedron. — Giovanni Italiano <giovanni.italiano.math@gmail.com>
A group is said to algebraically fibre if it admits an epimorphism onto $\mathbb{Z}$ whose kernel satisfies suitable finiteness properties, such as finite generation or finite presentability. The study of algebraic fibrations of hyperbolic groups has been a major theme in geometric group theory over the past two decades, motivated in part by the virtual fibering conjecture for odd-dimensional hyperbolic manifolds. Recently, Lafont, Minemyer, Sorcar, Stover, and Wells constructed, for every $d \geq 2$, a $d$-dimensional hyperbolic group admitting an algebraic fibration with finitely generated kernel. We strengthen this result by showing that, for every $d \geq 3$, there exists a $d$-dimensional hyperbolic group admitting an algebraic fibration whose kernel is _finitely presented_. Our construction combines right-angled Coxeter groups and Bestvina--Brady theory with a new collar-coning procedure inspired by a family of polytopes introduced by Löbell in the 1930s. This is joint work with M. Migliorini and A. Ng.
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An alternative for subgroups of Thompson group V — Corentin Bodart <cobodart123@gmail.com>
In this talk, I will explain some applications of the characterization of finitely generated subgroups of Thompson group V given in Davide Perego's talk. I will recall some of the previously known obstructions, start building up by rephrasing them in our new framework, and then give some ideas towards the following stronger alternative: every finitely generated subgroup of V is either virtually abelian, or contains a free non-abelian semigroup.
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Book decompositions and Morse complexity — Bena Tshishiku <bena_tshishiku@brown.edu>
A book decomposition of a manifold is a way of decomposing it as a union of codimension-1 submanifolds (the pages) that are glued along a codimension-2 submanifold (the binding). For example, each fibered knot gives a book decomposition of the 3-sphere; more generally, every odd dimensional manifold has a book decomposition by work of Alexander, Lawson, and Quinn. In this talk I’ll explain why even-dimensional hyperbolic manifolds do not have book decompositions. The main tool is Morse complexity, a norm on singular homology introduced by Gromov for which we give some new computations for locally symmetric manifolds. This is joint work with Fedor Manin and Shmuel Weinberger.
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- Plenary
Cohomology of arithmetic lattices and link complements — Jean Raimbault <jean.raimbault@univ-amu.fr>
I will present a proof of the following conjecture of Baker--Reid: given any rational homology 3-sphere N, there are at most finitely many congruence arithmetic quotients of hyperbolic space which are homeomorphic to the complement of a link in N. There are many ingredients to the proof but the final step is an asymptotic lower bound on the cuspidal homology of certain congruence subgroups of Bianchi groups. I will therefore use the conjecture as an excuse to talk about various ways to give such bounds, and finally present the somewhat new method we used to get to the result we needed. (Joint work with Steffen Kionke).
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Conjugacy separability in free-by-cyclic groups — Monika Kudlinska <mak74@cam.ac.uk>
A group G is conjugacy separable if any pair of non-conjugate elements remain non-conjugate in a finite quotient of G. While the original motivation for studying conjugacy separability stems from applications to algorithmic problems in group theory, more recently it has been successfully leveraged to exhibit certain rigidity properties of manifolds. In my talk, I will briefly discuss the applications of conjugacy separability to topology, before discussing the new result that all free-by-cyclic groups - a family closely related to 3-manifold groups - are conjugacy separable. This is joint work with Francois Dahmani, Sam Hughes and Nicholas Touikan.
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Convex cocompact groups with three-dimensional limit sets — Lorenzo Ruffoni <lorenzo.ruffoni2@gmail.com>
Given a discrete group of isometries of a real hyperbolic space, we can look at its limit set, i.e., the set of accumulation points of its orbits on the sphere at infinity. This is a compact metric space on which the group acts, and which enjoys many interesting geometric, topological, and dynamical features. For example, classical Kleinian groups provide many examples with limit sets that are quasicircles, Cantor sets, and Sierpinski carpets. In this talk, I will discuss how to construct examples whose limit sets are various low-dimensional trees of manifolds. In particular, we answer a question of M. Kapovich, by constructing convex cocompact groups of isometries of real hyperbolic spaces, whose limit sets are Čech cohomology 3-spheres not homeomorphic to S^3. These groups are right-angled Coxeter groups and our construction is flexible enough to produce infinitely many quasi-isometry classes. This is joint work with S. Douba, G.-S. Lee, and L. Marquis.
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Divergence in groups with microsupported action — Dominik Francoeur <dominik.francoeur@uam.es>
Divergence is a quasi-isometry invariant of groups that measures how difficult it is to connect two elements in the Cayley graph of a group by a path that does not pass close to the identity element. It is related to the existence of cut points in the asymptotic cones of the group. In this talk, we will explore divergence in groups with microsupported actions, a class of groups that include interesting examples such as Grigorchuk's groups and Thompson's groups. This is joint work with Letizia Issini and Tatiana Nagnibeda.
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Endomorphisms induced by self-maps in low dimensions — Christoforos Neofytidis <neofytidis.christoforos@ucy.ac.cy>
I will explain how residual finiteness and numerical invariants can be used to determine when all self-maps of non-zero degree induce an injective endomorphism or an automorphism of the fundamental group of a manifold in dimension three and in geometric settings in dimension four.
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Hurewicz-type formula for asymptotic dimension of countable approximate groups — Vera Tonić <vera.tonic@gmail.com>
In their theorem from 2006, A. Dranishnikov and J. Smith proved that if $f:G\to H$ is a group homomorphism, then the following formula for asymptotic dimension is true: $\mathrm{asdim} G \leq \mathrm{asdim} H + \mathrm{asdim} (\mathrm{ker} f)$. This result is known as the Hurewicz-type formula, after a 1927 theorem from classical topological dimension theory by W. Hurewicz, which inspired it. In this talk we will establish a similar formula to the one by Dranishnikov and Smith, for the following setup: whenever $(\Xi, \Xi^\infty)$ and $(\Lambda,\Lambda^\infty)$ are countable approximate groups and $f:(\Xi, \Xi^\infty) \to (\Lambda,\Lambda^\infty)$ is a (general) quasimorphism, i.e., a quasimorphism which need not be symmetric nor unital, then the following formula is true: $$ \mathrm{asdim} \Xi \leq \mathrm{asdim} \Lambda + \mathrm{asdim} (f^{-1}(f(e_\Xi)D(f)^{-1}D(f))), $$ where $D(f)$ is the defect set of the quasimorphism $f$. It follows as a corollary that if $f:G\to H$ is a quasimorphism of countable groups, then $$ \mathrm{asdim} G\leq \mathrm{asdim} H + \mathrm{asdim} (f^{-1}(f(e_\Xi)D(f)^{-1}D(f))).$$ In particular, whenever the quasimorphism $f$ is symmetric and unital, we can replace $f^{-1}(f(e_\Xi)D(f)^{-1}D(f))$ in the formulas above by $f^{-1}(D(f))$.
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- Plenary
Obstructing Riemannian smoothings on CAT(0) manifolds — Jean-François Lafont <jlafont@math.ohio-state.edu>
CAT(0) geometry is a metric generalization of Riemannian non-positive curvature. One could wonder, in the context of closed manifolds, if this is a genuine generalization? Up to dimension three, every closed manifold supporting a CAT(0) metric also supports a Riemannian non-positively curved metric. But this is no longer true when the dimension is >3. I will give an overview of the various known constructions of "exotic" CAT(0) manifolds in higher dimensions, culminating in a sketch of some new high dimensional examples (joint work w/ Bakul Sathaye).
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On the characteristic classes of hyperbolic manifolds — Stefano Riolo <stefano.riolo@unibo.it>
Every finite-volume hyperbolic manifold is finitely covered by a stably parallelizable manifold, which in particular has trivial Stiefel-Whitney and Pontryagin classes. On the other hand, it is difficult to produce hyperbolic manifolds with non-trivial characteristic classes. In two recent works, one with Rizzi and one with Bustamante and Reyes, we complement the previously existing results of Long-Reid, Martelli-R-Slavich and Chen on the theme, solving one of the K3 problems and answering questions of Charney-Davis and of Belegradek on strict hyperbolization, respectively. The talk will essentially consist of an overview on the subject.
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Profinite rigidity of Kähler groups — Claudio Llosa Isenrich <claudio.llosaisenrich@uni.lu>
A classical problem in complex algebraic geometry is understanding the topology of smooth complex projective varieties, and, more generally, of compact Kähler manifolds. Two natural topological invariants to consider are the fundamental group and its profinite completion; the latter is also known as the algebraic fundamental group. In this talk I will address the following questions: When is the fundamental group of a compact Kähler manifold uniquely determined by its profinite completion? And, when does the profinite completion even determine the homeomorphism type of the underlying manifold? In particular, I will explain positive answers to both questions in the case of a direct product of fundamental groups of closed hyperbolic Riemann surfaces. This talk is based on joint work with Hughes, Py, Stover and Vidussi.
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Rigidity for hyperbolic groups with Pontryagin sphere boundary — Emily Stark <emilyrstark@gmail.com>
The Pontryagin sphere is a homogeneous, nowhere planar, compact 2-dimensional fractal constructed as an inverse limit of closed orientable surfaces. The Pontryagin sphere arises naturally as the boundary at infinity of the fundamental group of a 3-dimensional hyperbolic pseudo-manifold. We prove that if the conformal dimension of the boundary is less than four, then such a group is action rigid: if it acts geometrically on the same proper metric space as another group, then the groups are virtually isomorphic. A key component of the proof is a generalization of Yang's Theorem regarding the structure of p-adic actions on a tree of manifolds. This is joint work with Chris Cashen, Pallavi Dani, and Kevin Schreve.
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Subgroups of Coxeter groups and Stallings Foldings — Jake Murphy Murphy <jmurphy4@oberlin.edu>
Stallings introduced the concept of Stallings foldings to aid in the study of free groups, which creates a "folded graph" associated with a subgroup of a free group. Dani-Levcovitz adapted this concept to the setting of Right-Angled Coxeter Groups. In this talk, we will generalize this idea to certain finitely generated subgroups of Coxeter groups to determine their index, whether they are normal, and to find generating sets of their intersections.
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Unipotents and linearity of amalgams — Sami Douba <doubasami@gmail.com>
I will discuss joint work with Konstantinos Tsouvalas where we investigate linearity of amalgams of subgroups of algebraic groups along intersections with algebraic subgroups. In the process, we establish linearity of certain “doubles” of linear groups, and obtain new examples of finitely generated residually finite groups that fail to be linear.
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Vanishing of bounded cohomology beyond amenability — Caterina Campagnolo <caterina.campagnolo@uam.es>
Bounded cohomology is an invariant of groups and spaces developed by Gromov in the 80's. Despite its purely topological definition, it turns out to have deep relations with geometric properties of the spaces or algebraic properties of the groups under consideration. In particular, its vanishing for a large family of coefficients modules allows to characterize amenability. In joint work with Fournier-Facio, Lodha and Moraschini, we develop a new criterion for the vanishing of bounded cohomology for a subfamily of coefficients modules and apply it to a variety of examples of groups of topological, geometric and dynamical origin. We also remark an interesting relationship with homological stability of these families of groups.
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