(joint work with A.Rasmussen and C.Abbott) Recent papers of Balasubramanya and Abbott-Rasmussen have classified the hyperbolic actions of several families of classically studied solvable groups. A key tool for these investigations is the machinery of confining subsets of Caprace-Cornulier-Monod-Tessera, which applies in particular to solvable groups with virtually cyclic abelianizations.
In this talk, I will explain the motivation behind our research and the classification of hyperbolic actions in the known cases so far. I will then talk about recent and ongoing work that explores generalizations of the techniques used to achieve these results.
Largely due to its applications to $p$-groups, understanding the intersection of a set of classical groups has long been a problem of interest in computational algebra. It reduces to the following computational problem: given a set of reflexive forms on a common vector space, construct generators for the group of invertible linear transformations that preserve every form in the set.
The first effective algorithms to solve the problem were developed for specific pairs of forms in joint work with E.A. O'Brien as long ago as 2008. More generally, by viewing the individual forms instead as a single bilinear map, the target group of isometries can be realized as the group of unitarian elements of an algebra with involution. This perspective was exploited in a joint 2012 paper with J.B. Wilson to develop a polynomial-time algorithm to solve the isometry group problem over finite fields of odd characteristic.
Unfortunately, the approach taken in the 2012 paper founders over fields of characteristic 2 in two significant ways; despite continued interest in the problem, this case has remained open for the past decade. In this talk I will present some recent developments for the characteristic 2 case. This is a report on joint work with M. Kassabov and J.B. Wilson.
For positive integers $r$, $s$ and $t$, the ordinary $(r,s,t)$ triangle group $\Delta(r,s,t)$ is the abstract group with presentation $\langle\, x,y,z \, | \, x^r = y^s = z^t = xyz = 1\,\rangle$. Such groups play a role in the study of regular maps on surfaces, and compact Riemann surfaces and algebraic curves of genus $g > 1$ with large automorphism groups.
In a 2016 paper by Alan Reid, Marston Bridson and the speaker, it was shown using the theory of profinite groups that if $\Gamma$ is a finitely-generated Fuchsian group and $\Sigma$ is a lattice in a connected Lie group, such that $\Gamma$ and $\Sigma$ have exactly the same finite quotients, then $\Gamma$ is isomorphic to $\Sigma$. As a consequence, two triangle groups $\Delta(r,s,t)$ and $\Delta(u,v,w)$ have the same finite quotients if and only if $(u,v,w)$ is a permutation of $(r,s,t)$.
A direct proof of this property of triangle groups was given in the final section of that paper, with the purpose of exhibiting explicit finite quotients that can distinguish one triangle group from another. Unfortunately, part of the latter direct proof was flawed. In this talk I'll describe two new direct proofs, one being a corrected version using the same approach as before (involving direct products of small quotients), and the other being a shorter one that uses the same preliminary observations as in the earlier version but then takes a different direction (involving further use of the ‘Macbeath trick’).
I will present a survey of some recent work with Bernd Schulze that generalises two well known results in combinatorial geometry to a symmetric setting. One concerns the graphs that can arise as contact graphs of collections of segments in the plane and the other concerns pseudo-triangulations of the plane. A pseudo-triangle is polygon that has exactly 3 convex internal angles.
I will discuss the growth of the number of infinite dihedral subgroups of lattices in $\mathrm{PSL}(2,\mathbb{R})$. Such subgroups exist whenever the lattice has $2$-torsion and they are closely related to so-called reciprocal geodesics on the corresponding quotient orbifold. These are closed geodesics passing through the order two orbifold point, or equivalently, homotopy classes of closed curves having a representative in the fundamental group that's conjugate to its own inverse. We obtain the asymptotic growth of infinite dihedral subgroups (or reciprocal geodesics) in any lattice, generalizing earlier work of Sarnak and Bourgain-Kontorivich on the growth of the number of reciprocal geodesics on the modular surface. Time allowing, I will explain how our methods also show that reciprocal geodesics are equidistributed in the unit tangent bundle. This is joint work with Juan Souto.
The geometric classification of 2- and 3-dimensional closed manifolds stands as one of the most remarkable achievements of modern mathematics. The spectacular proof of Thurston's geometrization conjecture in the 3-dimensional case by Perelman twenty years ago naturally leads to the possibility that similar successes await in dimension 4. In this talk I will discuss the relative weakness of such methods in dimension 4 and argue that, for a variety of reasons, new geometric ideas are desperately needed. At the heart of these difficulties is the current mess that is 4-manifold topology and the absence of geometric paradigms of sufficient strength to accommodate the extra dimension.
A classical graph invariant is the girth, which is the length of the shortest cycle. In the presence of weights or distances assigned to the edges, one can similarly define the weighted girth or systole of a graph. Bollobás and Szemerédi have proved asymptotic bounds on this quantity as the graph Betti number goes to infinity. I will discuss new bounds for the case of small Betti number proved recently in joint work with Michael Wiemeler and Burkhard Wilking. This has implications for structure of torus representations with connected isotropy groups and applications to the problem of classifying Riemannian manifolds with positive curvature and large isometry groups.
In this talk I will present some recent results obtained in collaboration with G. Carron and D. Tewodrose about the structure of Gromov-Hausdorff limits of manifolds with Ricci curvature satisfying a Kato integral bound. We obtain similar results to the one of the Cheeger-Colding theory for the case of a lower Ricci bound: after explaining the context and motivation of this work, I will focus on almost rigidity results, such as the stability of the torus in case the first Betti number is equal to the dimension.
Let $V$ be a representation of a group $G$ in characteristic 0. Even if $V$ is irreducible its reduction modulo $p$ is in general not irreducible and often not even homogeneous, that is it has non-isomorphic composition factors.
Given a group $G$ a natural question is to characterise irreducible representation which remain irreducible or homogeneous when reduced to characteristic $p$.
In this talk I will present reduction results on the classification of (almost) homogeneous reductions of spin representations of symmetric groups in characteristic 3. From this result it follows that, in characteristic 3, homogeneous reductions of spin representations of symmetric or alternating groups are actually irreducible.
This is joint work with Matthew Fayers.
Given an orthogonal representation of a Lie group $G$ on a Euclidean vector space $V$, Invariant Theory studies the algebra of $G$-invariant polynomials on $V$. This setting can be generalized by replacing the representation $G$ with a map from $V$ to a metric space $X$, with smooth and equidistant fibers, which mimics the projection onto the orbit space. In this case, one can study the algebra of polynomials that are constant along the fibers - thus producing an Invariant Theory, but without groups. In this talk we will discuss a surprising relation between the geometry of these maps and the corresponding “invariant” algebra, including recent joint work in progress with Ricardo Mendes and Samuel Lin, showing how to estimate volume and diameter of the base $X$ using the algebra.
The notion of a growth competition between two deterministically growing clusters in a complete, non-compact metric space (or graph) was first proposed by I. Benjamini and recently explored in the case of 2-dimensional Euclidean and hyperbolic spaces by his student, R. Assouline. A growth competition in a non-compact, complete Riemannian manifold, $X$, (or more generally a complete, non-compact geodesic metric space) is the existence of two sets, $A_t$ (fast) and $B_t$ (slow), $t \geqslant 0$, that grow from singletons according to the following simple rules:
A key result shown by Assouline is that given any two distinct points $p,q$ in a path connected, complete, geodesic metric space $X$ and a real number $\lambda >1$, there exists a unique growth competition satisfying the above conditions. A basic geometric question one may ask in this setting is: Under what circumstances is the slow set, $B_\infty$, totally bounded (surrounded) by the fast set, $A_\infty$, versus when are they both unbounded (co-existence)? The applications of this geometric exploration are evident in a variety of settings (including disease/vaccine vectors, flow of misinformation or the control of forest fires). In recent work with Benjamin Schmidt and Ralf Spatzier we have been exploring the above question in the setting of non-positive curvature. In this talk we introduce growth competitions and give a preview of some results and open problems.
For a prime number $p$, an elliptic $p$-group is a group of class $2$ and exponent $p$ whose Pfaffian describes an elliptic curve. I will report on joint work with Christopher Voll and ongoing work with Joshua Maglione on the computation of autmorphisms and isomorphism classes of elliptic groups, leveraging on the geometry of the defining elliptic curve.
You can define a nilpotent group from the pixels of a digital movie, which just shows that these groups are extremely varied and defy tidy classifications and structure theorems. Even so, recent advances are being made by extracting geometries from nilpotent groups and turning to classical geometric methods. This talk will explore the influence of 20th century “Coordinatization Theorems” on 21st century counting problems in Group Theory.
Reports on solo and joint work with Martin Kassabov, Cornell.
In addition to the classical notions of scalar, Ricci and sectional curvature in Riemannian geometry, there are also natural notions of curvature which interpolate between these. Although there is relatively little in the literature concerning these measures of curvature, studying intermediate curvatures offers a more nuanced view of curvature in Riemannian geometry. Indeed, one might hope to arrive at an enhanced understanding of the classical curvatures in this way.
In the first part of the talk, I will define intermediate curvatures, and discuss how these have arisen in the literature both historically, and more recently. In the second part of the talk I will describe some joint work which explores the connections between intermediate curvatures and topology.