GAGTA 2026

Let $G$ be a rank 2 Kac–Moody group over a finite field $k$, and let $X$ denote its associated Tits building. The geometric and combinatorial structure of $X$ plays a central role in understanding analytic properties of arithmetic subgroups of $G$. In earlier work, Ali and Carbone used the action of subgroups associated with the negative BN-pair, namely $U^{-}$, $B^{-}$, and $P_1^{-}$, to show that these groups arise as lattice subgroups of $G$. In this talk, we focus on the standard parabolic subgroup $\Gamma =P_1^{-}$ and introduce Eisenstein series on the quotient space $\Gamma \backslash X$. Using the geometry of the building together with the Iwasawa decomposition of the Haar measure on $G$, we establish convergence of these Eisenstein series in a suitable half-space and prove their meromorphic continuation. The talk will also discuss how these methods relate to earlier lattice constructions of Ali and Carbone, as well as possible extensions to higher-rank Kac–Moody groups and related geometric settings. This is joint work with Lisa Carbone and Paul Garrett.

I will present a new notion of non-positively curved groups: the collection of discrete countable groups acting (AU-) acylindrically on finite products of hyperbolic spaces. This work (joint with T. Fernos) is inspired by the classical theory of S-arithmetic lattices and that of acylindrically hyperbolic groups. I will start with the motivation for studying these groups in this talk, and cover some results we have proved that highlight this duality.

Erdős and Rényi introduced a model for studying random graphs of a given density and proved that there is a sharp threshold at which lower density random graphs are disconnected and higher density ones are connected. This framework is extremely well-suited for studying right-angled Coxeter groups (RACGs) and we will describe some theorems analogous to the Erdős–Rényi result in this context. Moreover, in trying to understand random groups, we have discovered some surprising new results, including for instance a simple constraint on a presentation graph which forces the associated RACG to be relatively hyperbolic.

I will discuss an approach based on boundary representations to establish an optimal form of property RD for discrete groups acting properly and cocompactly on affine buildings of type $\tilde{A}2$. This improves a celebrated result of Robertson, Ramagge, and Steger from the late 1990s establishing property RD, and is related to the conjecture of Kazhdan and Yom Din concerning the decay of matrix coefficients of unitary representations. After a brief introduction to property RD, I will emphasize geometric arguments involving the Furstenberg boundary, in particular the notion of foldings (or "confluences," following V. Kaimanovich), together with tools from harmonic analysis on the boundary of buildings.

The dynamical properties of the geodesic flow on a negatively curved manifold tell us about the underlying geometric structure of the manifold. Using (Hölder) reparameterizations (i.e., time changes) of the flow it is possible to encode information about multiple geometries within a single dynamical framework. This allows one to compare different geometries and to obtain results of various flavours: counting results, limit laws, rigidity results, etc. In this talk we will explore the space of continuous reparameterizations of geodesic flows. We present a coarse geometric description of these reparameterizations that sheds light on their structure and rigidity. Joint work with D. Martinez-Granado and E. Reyes.

We discuss how Steinberg groups appear in the construction of groups for infinite dimensional Lie algebras, namely a Lie group analog for the Monster Lie algebra.

Smale spaces were introduced in the late 1970s by David Ruelle in his influential monograph on thermodynamic formalism. These dynamical systems include: Anosov diffeomorphisms, non-wandering sets of Smale's Axiom A diffeomorphisms, various types of solenoids and attractors, as well as (in a symbolic dynamical setting) subshifts of finite type. In a recent joint work with Michel Coornaert, we proved, among other things, a Garden of Eden type theorem (GOET) for irreducible Smale spaces. This generalizes previous results by Fiorenzi (GOET for irreducible subshifts of finite type) and ours (GOET for Anosov diffeomorphisms on tori).

The quantitative behavior of random walks on groups is rather well-understood for hyperbolic groups and their generalizations. However, even the most basic questions become challenging if the geometry of the underlying group is sufficiently different from the geometry of negative curvature. In particular, multiple questions remain open for the wreath products of finitely generated groups. In this talk, we will discuss a Central Limit Theorem for the drift of a non-elementary random walk on a wreath product with an acylindrically hyperbolic base group. Moreover, we will also provide tight upper bounds on the central moments of the drift in this setting. The talk is based on joint work with Christian Gorski and Eduardo Silva.

I will review some classical examples of groups whose strictly descending chains of subgroups are finite. Some recent constructions of such groups (having large cardinality) will also be explained. Set-theoretic consistency will play a role in some of the stated results. This is joint work with Saharon Shelah.

I will explain an attempt to give an analytic formulation of Thurston's minimal stretch maps and geodesic laminations. This is joint work with Karen Uhlenbeck.

We give a complete criterion for when two hyperbolic automorphisms of a tree generate a free, discrete subgroup. The decision depends only on three geometric invariants: the translation lengths of the generators and the length of overlap of their axes. This data is organized using the continued-fraction expansion of the translation-length ratio.

A group is said to be "defined by its types" provided it is determined up to isomorphism by the set of types realized by the tuples of its elements. For finitely generated groups $G$, a sufficient condition to be defined by its types is that every elementary embedding $G\to G$ be an automorphism. Finitely generated $G$ satisfying the above condition are said to be "strongly defined by types." In Myasnikov and Romanovskii (2018), "Characterization of finitely generated groups by types," it was proven that, with the possible exception of the non-orientable surface groups of genus 2 and 3, every surface group (orientable or non-orientable) is strongly defined by its types. In this talk, I report on joint work with G. Rosenberger and D. Spellman which closes that gap. In particular, the non-orientable surface groups of genus 2 and 3 are strongly defined by types. Indeed, the non-orientable surface group of genus 3 (but not the non-orientable surface group of genus 2) satisfies the stronger property of being cohopfian.

It is a longstanding open problem whether all one-relation monoids have decidable word problem. Important results of Ivanov, Margolis, and Meakin (2001) show that this problem can be reduced to solving the word problem in certain families of one-relator inverse monoids. Their results also show that the word problem for one-relator inverse monoids can often be reduced to solving the membership problem in certain submonoids of one-relator groups. This motivates the study of the membership problem in submonoids of one-relator groups. In this talk I will speak about recent joint work with Islam Foniqi in which we prove several decidability and undecidability results about the submonoid membership problem in one-relator, surface and hyperbolic groups. The membership problem in free-by-cyclic one-relator groups is undecidable in general. I will present a general result that gives sufficient conditions under which membership in submonoids of certain free-by-cyclic one-relator groups is decidable, and I will explain how we can apply this result to show that membership in various families of submonoids of surface groups is decidable. I will discuss graded submonoids of one-relator groups, presenting a result that gives conditions under which such a submonoid has decidable membership with linear distortion. These results are then applied to improve on a result of Margolis, Meakin and Sunik by showing the prefix monoid of a surface group has linear distortion. I will explain how these results can be used to prove the Magnus submonoid membership problem is decidable in several families of one-relator groups. Here a Magnus submonoid of a one-relator group is one generated by a strict subset of the generators and their inverses. I will also present some results about the related positivity problem (also called the quasi-Magnus problem) which asks whether membership in the positive submonoid generated by the generators is decidable. In particular I will give a solution to a problem posed by McCammond and Meakin in 2006 which asked whether this problem is decidable in hyperbolic groups. This is joint work with Islam Foniqi.

Consider an infinite word over a finite alphabet (or, more generally, a subshift). A quantitative measure of its combinatorial complexity is given by the complexity function, which counts, for each natural number $n$, the number of distinct subwords of length $n$ that occur in it. This notion plays a fundamental role in dynamical systems, combinatorics on words, computer science, Diophantine analysis, and beyond. We solve the inverse problem of determining which functions can arise — up to bi-Lipschitz equivalence — as complexity functions of infinite words and subshifts. Further illustrating the richness of the variety of complexity functions, we construct infinite words whose complexity functions are arbitrarily close to linear, but whose discrete derivatives exhibit exponential-type behavior, thereby resolving in a strong sense a problem posed by Cassaigne in 1995. As another application of our constructions, we exhibit infinite words of polynomial complexity whose Rauzy graphs do not admit Benjamini–Schramm limits. The talk is based on joint works with Leeman, Moreira, and Zelmanov.

In my talk, based on two joint works with Yaroslav Vorobets, I will give a quick introduction to ample (or topological full) groups associated with group actions on a Cantor set and then present a result about embedding of one of the groups of intermediate growth into a topological full group associated with the Thue–Morse substitution. I will then switch to the matter of classification of maximal subgroups in ample groups and present several results in this direction.

The 1-skeleton of an $\tilde{A}2$ Bruhat–Tits building is isomorphic to the Cayley graph of an abstract group with relations coming from "triangle presentations." This abstract group either embeds into $\mathrm{PGL}(3,{𝔽}_q(\left(x\right)\left)\right)$ or $\mathrm{PGL}(3,{\mathbb{Q}}_q)$, or else is exotic. Currently, the complete list of triangle presentations is only known for projective planes of orders $q=2$ or $3$. However, one abstract group that embeds into $\mathrm{PGL}(3,{𝔽}_q(\left(x\right)\left)\right)$ for any prime power $q$ is known via the trace function corresponding to the finite field of order $q^3$. I found a new method to derive this group via perfect difference sets. This method demonstrates a previously unknown connection between difference sets and $\tilde{A}2$ buildings. Moreover, this method makes the final computation of triangle presentations easier, which is computationally valuable for large $q$.

The actions $x\mapsto 2x\left(\mathrm{mod}1\right)$ and $x\mapsto 3x\left(\mathrm{mod}1\right)$ on the circle are very different, although both preserve Lebesgue measure. Each also preserves many other non-atomic probability measures on the circle. In this talk, I will present several examples of invariant measures with different geometric types. Beginning with Furstenberg's pioneering work, progress was made in the study of the $2\times$ and $3\times$ actions, but many problems remain open. A central question is whether Lebesgue measure is the only common non-atomic probability measure invariant under both actions. Furstenberg conjectured that Lebesgue measure is the only common non-atomic ergodic probability measure invariant under both actions. Rudolph proved this conjecture under the additional assumption of positive measure-theoretic entropy. In the 1980s, Sullivan introduced several ideas from the boundary actions of Kleinian groups into the study of one-dimensional dynamical systems, including bounded geometry, quasisymmetry, and quasiconformality. I will explain how these ideas can be applied to the study of the $2\times$ and $3\times$ actions on the circle. In particular, I will introduce a balanced geometry condition and prove that Lebesgue measure is the only common probability measure with balanced geometry invariant under both actions. This result holds more generally for the actions $x\mapsto px\left(\mathrm{mod}1\right)$ and $x\mapsto qx\left(\mathrm{mod}1\right)$ whenever $(p,q)=1$.

Motivated by a classic theorem of Birman and Series about the set of complete simple geodesics on a hyperbolic surface, we study the Hausdorff dimension of the set of endpoints in $\partial F_r$ of some abstract algebraic laminations associated with free group automorphisms. For an exponentially growing outer automorphism $\varphi \in Out\left(F_r\right)$ we show that the set of endpoints $E_L\subseteq \partial F_r$ of any of the attracting laminations $L$ of $\varphi$ has Hausdorff dimension 0 and packing dimension 0, for any visual metric on the boundary $\partial F_r$, and similarly that $L\subseteq {\partial }^2{F}_r$ (where ${\partial }^2{F}_r$ is equipped with the product metric of a visual metric) has Hausdorff dimension 0 and packing dimension 0. If $\varphi \in Out\left(F_r\right)$ is atoroidal and fully irreducible, we deduce the same conclusions for the ending lamination ${\Lambda }_{\varphi }$ of $\varphi$ that gets collapsed by the Cannon–Thurston map $\partial F_r\to \partial G_{\varphi }$ for the associated free-by-cyclic group $G_{\varphi }=F_r{⋊}_{\varphi }\mathbb{Z}$. By contrast, the set of endpoints of any of these laminations has upper box dimension $>0$ for any visual metric on $\partial F_r$.

Perelman's proof of the Poincaré conjecture shows that every simply connected closed 3-manifold is homeomorphic to the 3-sphere. The fundamental groups of 3-manifolds attract lots of interest from mathematicians of different fields. As it was stated in a famous survey of Allen Hatcher, "The classification of 3-manifolds," one would want to know exactly which groups occur as fundamental groups of these manifolds. The Stallings–Jaco–Hempel reformulation of the Poincaré conjecture inspired several connections between low-dimensional topology, equations over free groups, and combinatorial group theory. The reformulation reduces the problem to study epimorphisms from the fundamental group of a closed orientable surface onto the direct product of two free groups (they correspond to Heegaard splittings of 3-manifolds and were named splitting homomorphisms). Olshanskii (1989) constructed (in non-explicit form) first non-trivial examples of such splitting epimorphisms and verified the standardness of some of them. We construct up to equivalence all the splitting coordinate-surjective homomorphisms (among them, the genuine splitting epimorphisms are exactly those for which our constructed associated group balanced presentation is trivial). We give generators and relations of the corresponding balanced presentation (so all closed orientable 3-manifold groups) that can be studied by algebraic methods. We also analyse a large class of such homomorphisms/presentations (including all Olshanskii's epimorphisms) and show that splitting epimorphisms are very rare and in this case they are standard. There are two main motivations of this work. One is the real possibility of obtaining a purely group-theoretic proof of the Poincaré conjecture. Another is to study fundamental groups of closed orientable 3-manifolds in Hatcher's spirit that become clear from their presentations that we construct. This is joint work with Alina Vdovina.

For a free group $F_r$, let $S_n$ be the set of all freely reduced words of length $n$ and $B_n$ be the set of all words of length at most $n$. $S_n$ and $B_n$ are the sphere and the ball, respectively, of radius $n$ in the Cayley graph of $F_r$. We will use Stallings foldings to uncover with simple graph theory some surprising properties about the ranks of subgroups generated by elements of $S_n$ or $B_n$. In particular, we will find the maximum possible rank of such subgroups, thereby answering a question posed by V. Dotsenko, published as Problem F42 in Open Problems on Infinite Groups by Kapovich, Myasnikov, and Shpilrain.

The theory of one-relator groups was initiated almost a century ago when Magnus proved the Freiheitssatz. Until very recently, the theory remained in the realm of combinatorial group theory. In this talk I will survey the recent advances which instead have come from homological and geometric group theory and discuss some future directions.

Consider a cover of the Riemann sphere by a compact connected Riemann surface. The monodromy group of the cover is an important invariant describing how badly the cover degenerates. It is natural to ask which groups can appear in such a context. We will survey some applications of this question and discuss how the genus of the surface influences the answer. In joint work with Danny Neftin and Spencer Gerhardt, we give an answer for groups of Aschbacher–Scott type B in genus at most 1.

Stix and Vdovina used quaternion algebras over function fields to construct explicit families of torsion-free arithmetic lattices acting simply transitively on products of Bruhat–Tits trees of equal regularity. In particular, they obtained a lattice of Euler characteristic one over ${𝔽}_3\left(t\right)$ acting on a product of two trees. In this talk, I will discuss analogous constructions over number fields. I will describe classification results for lattices of small Euler characteristic and present new infinite families of torsion-free lattices acting simply transitively on products of trees of equal regularity. This presentation contains joint work with Jakob Stix, Alina Vdovina, and Jiahui Yu.

We discuss the regularity theory of harmonic maps into Euclidean buildings, without assuming local finiteness. The main result shows that such maps have singular sets of Hausdorff codimension at least 2, extending the regularity theorem of Gromov and Schoen for locally finite buildings. As an application, we establish a rank-one superrigidity theorem for algebraic groups over fields with non-Archimedean valuations. This generalizes the rank-one superrigidity theorem of Gromov and Schoen for $p$-adic groups and places the higher-rank superrigidity theorem of Bader and Furman in a geometric framework.

GAGTA is 20 years old. In this talk I will discuss some ideas and research areas that captivated our imagination from 20 years ago and up till now. What looked promising and how it aged, what problems were solved and which ones stay open still. All of these are through the prism of GAGTA conferences, books, and participants.

I will discuss higher-dimensional versions of property (T) in the form of vanishing of higher cohomology with coefficients in unitary representations. I will prove a new, algebraic characterization of vanishing of such cohomology in terms of sums of squares. This requires a new correspondence between higher cocycles and positive functionals. In particular, I will discuss a higher-dimensional generalization of the classical Gelfand–Naimark–Segal construction. This is joint work with Antonio Lopez Neumann.

I will present a procedure that turns certain abstract presentations, such as those for surface groups, Thompson groups, and Artin groups, into topological groups with a prescribed open subgroup. These all share some topological properties depending on the chosen open subgroup. For studying cohomological finiteness conditions, we concentrate on totally disconnected locally compact RAAGs and present a generalisation of the Salvetti complex for these groups. This is joint work with Ilaria Castellano, Bianca Marchionna, and Yuri Santos Rego. This talk will be via ZOOM.

Let $u$ and $v$ be cyclic words in a free group and suppose they have the same length. We want to know when $u$ and $v$ continue to have the same length under every free group automorphism. We call words with this property translation equivalent. I'll give some necessary conditions and some sufficient conditions for a pair of words to be translation equivalent based on simple combinatorial observations about the words. Then I'll describe a new, fast algorithm for deciding translation equivalence based on those criteria. Translation equivalence in free groups was introduced by Kapovich, Levitt, Schupp, and Shpilrain in analogy to hyperbolic equivalence of curves on surfaces. I'll briefly explain this analogy and give an application of translation equivalence for curves on surfaces. Finally, I'll discuss a generalization of this property called bounded translation equivalence, which was introduced by the same authors.

20 years ago, the seminal work of Polterovich and Sodin showed a short but surprising result: if the iterates of derivatives of a diffeomorphism of the interval of class $C^2$ grow subexponentially, then they grow at most quadratically. In this talk we will discuss a stronger result asserting that $\underset{n\to \mathrm{\infty }}{lim}{max}_x\frac{D{f}^n\left(x\right)}{n^2}$ exists. We briefly discuss the modern tools that allow us to obtain these results. Joint work with Andrés Navas.

The spectral gap of the Laplacian on a hyperbolic surface measures how well the surface is connected. It was shown long ago by Huber that the spectral gap of such a surface cannot exceed that of the hyperbolic plane, asymptotically as the genus goes to infinity. Whether there exists a sequence of closed hyperbolic surfaces that achieves this bound — an old conjecture of Buser — was settled a few years ago by Hide and Magee. This was done by exhibiting a sequence of finite covering spaces of a fixed base surface that have good spectral properties. In this talk, I will discuss joint work with Magee and van Handel where we show that this phenomenon is in fact much more prevalent: given any closed hyperbolic surface, not only do there exist finite covers that have good spectral properties, but this is in fact the case for all but a vanishing fraction of its finite covers. The proof hinges on new developments on the notion of strong convergence in random matrix theory. I plan to give a gentle introduction to this line of works.

Toroidal solenoids and ${\mathbb{Z}}^d$-odometers provide a rich class of dynamical systems arising from actions of abelian groups on compact spaces with strong arithmetic structure. In this talk, I will discuss recent results connecting the classification and rigidity of such systems with arithmetic properties of integer matrices and associated number-theoretic invariants.

We explore parallels between the theory of letter-to-letter transducers and Mealy automata on one end and the theory of Latin squares on the other. We characterize various classes of automata (Mealy, reversible, invertible, bireversible) in terms of combinatorial structures analogous to those associated with Latin squares. In particular, we represent the inversion and dualization of transducers as analogues of parastrophies of Latin squares, and show that the isotopies of the quasigroups associated to Latin squares generalize the notion of isomorphism of transducers, preserving the class of bireversible automata.

It is well-known that, for mixing subshifts of finite type, weighted averages of periodic orbital measures become equidistributed with respect to the Gibbs measure associated to the weighting. The same holds for countable state shifts under more restrictive assumptions (both on the shift and on the weighting potential). One can ask the same question for periodic orbits in group extensions of these systems by finitely generated amenable groups. It turns out that one still has equidistribution but, in general, to a different Gibbs measure. This measure has a thermodynamic characterization. A key point is equality of Gurevich pressure for the skew product and its abelianization, which generalizes results about the spectral radii of random walks. Some of this is joint work with Rhiannon Dougall.

We study the rate distortion dimension of Gibbs measures for the full one-sided shift over a compact metric space. This is a natural extension of the classical shift on finite alphabets. However, when the alphabet is infinite, like the unit interval, the Kolmogorov–Sinai entropy diverges in general. To overcome this, we use rate distortion dimension, which is known to be related to mean dimension via the double variational principle. We assume that the compact metric space is Ahlfors regular and consider a Lipschitz potential depending only on the first two coordinates. We show that the rate distortion dimension of the corresponding Gibbs measure coincides with the Hausdorff dimension of the alphabet.

We will discuss connections between the asymptotic behavior and abstract algebraic properties in some classes of finitely generated and finitely presented semigroups including inverse semigroups, with a focus on the polynomial growth case. Part of the talk is joint work with Stuart Margolis.

The problems that we consider in this talk are mostly concerned with random products of given matrices over the reals. This includes the following questions: What is the expected absolute value of the largest (in absolute value) entry in such a random product? What is the (maximal) Lyapunov exponent for a random matrix product like that? We give an answer to the first question under some mild restrictions on the entries of the given matrices. For the second question, we offer a very simple and efficient method to produce an upper bound on the Lyapunov exponent. This part of the talk is based on joint work with N. Nabahi. In conclusion, we will discuss a related problem of complexity of multiplying matrices over rationals. This part of the talk is based on joint work with A. Olshanskii.

We construct random covers of higher-rank locally symmetric spaces, which we call scaffolded Poisson processes. Using these covers, we prove quantitative bounds, in terms of the injectivity radius, on the minimal number of generators of higher-rank lattices and on the size of the torsion in first homology. We then show that for Benjamini–Schramm convergent sequences of higher-rank lattices, the normalized growth of these invariants vanishes. This confirms the degree-one vanishing predicted by a conjecture of Bergeron and Venkatesh, and answers questions of Abért, Gelander, and Nikolov.

Systolic complexes were introduced by Januszkiewicz and Świątkowski to provide a simplicial analogue of non-positive curvature. A systolic group is one that acts geometrically on such a complex, inheriting robust geometric and algebraic properties including biautomaticity and the resulting solvability of the word and conjugacy problems. In joint work with Bob Gilman and Alden Walker, we provide a general framework for determining whether a finitely presented group $G=⟨S\mid R⟩$ is Cayley systolic, meaning that the flag completion of its Cayley graph $\Gamma (G,S)$ is systolic. This approach builds on results of M. Soergel as well as prior work of Gilman. Our methods offer a streamlined proof of the known result that triangle- and square-free Artin groups, as well as many Garside groups, are systolic.

We study one-variable equations over the lamplighter group $L_2={\mathbb{Z}}_2\wr \mathbb{Z}$. While the decidability of arbitrary equations over $L_2$ remains open, we prove that the Diophantine problem for single equations in one variable is decidable. Our approach reduces the problem to a divisibility question for families of parametric Laurent polynomials over ${\mathbb{Z}}_2$, whose exponents depend linearly on an integer parameter. To analyze this divisibility problem, we introduce a symbolic division procedure for associated bivariate polynomials and derive explicit bounds on the parameter from the structure of the resulting quotient and remainder. This yields an explicit decision procedure with exponential worst-case complexity. On the other hand, we show that for a generic class of equations, solvability can be decided in nearly quadratic time. Joint work with Yankun Wang.

The conjugator length function of a finitely generated group is the function $f$ so that $f\left(n\right)$ is the minimal upper bound on the length of a word realizing the conjugacy of two words of length at most $n$. This function provides a measure for the complexity of a direct approach to the Conjugacy Problem for the finitely generated group. I will discuss what functions can be realized as the conjugator length function of a finitely presented group and the connection of this function with other important invariants of finitely presented groups.

In this talk, we discuss the first theoretical framework guaranteeing that computers can, in principle, be used to analyze the parameter space of complex Hénon maps. More precisely, we obtain computability results for hyperbolic polynomial diffeomorphisms of ${ℂ}^2$, for which Hénon maps are prototypical examples. Specifically, we establish computability of the Julia set for hyperbolic maps, semi-decidability of hyperbolicity, and lower computability of the hyperbolicity locus in the parameter space of generalized Hénon mappings of fixed degree at least two. We also discuss extensions of our results to more general classes of hyperbolic systems. The results presented in this talk are joint work with Suzanne Boyd.

Mapping classes on graphs and surfaces give rise to flows or semiflows on their mapping tori, and taking limits for a sequence of such flows or semiflows might result in end periodic mapping classes, which are proper mapping classes on infinite type graphs and surfaces that have many similarities with the mapping classes on finite surfaces and graphs. I will discuss my prior work with Yan Mary He, Marissa Loving, and Paige Hillen on the normal form, symbolic coding, and entropy calculation of these mapping classes (arXiv:2408.13401, 2501.17389, and 2603.20491), and the ongoing work on relating them with the Veering polynomials of 3-manifolds.

A pattern in a group consists of the left cosets of a collection of subgroups. A quasi-isometry between two groups with patterns is pattern-preserving if it coarsely permutes the left cosets in the patterns. There are two natural questions concerning PPQIs (pattern-preserving quasi-isometries). The first is to determine when there is a PPQI between two groups with patterns. The second is whether PPQIs exhibit rigidity properties. We will discuss the second question in the setting of nilpotent Lie groups. We show that every self PPQI is at finite distance from an automorphism if the subgroups intersect the center trivially and generate the whole group and one of the following holds: (1) $G$ is a 2-step simply connected nilpotent Lie group; (2) $G$ is a free nilpotent Lie group. This is ongoing joint work with Mitra Alizadeh, Hao Liang and Qingshan Zhou.