arXiv subjects

Thomas Koberda

Publications and source records attributed to Thomas Koberda.

At least 19 recordsLinked to original sources

arXiv2026

Training Mathematicians in the Age of AI: Intellectual Agency, Cognitive Offloading, and the PhD Thesis

Powerful artificial intelligence is weakening the traditional relationship between mathematical output and evidence of mathematical expertise. In particular, the production of an original theorem or a polished dissertation can no longer, by itself, certify the intellectual formation of its nominal author. I argue that graduate mathematical education should therefore be organized around the formation of intellectual agency: internal technical competence, mathematical judgment, understanding, and responsible participation in a shared intellectual culture. I distinguish productive from premature cognitive offloading, propose complementary independent and AI-augmented modes of training, and suggest a corresponding reformulation of the role of the PhD thesis and dissertation defense. More broadly, I argue that academic mathematics should understand itself increasingly as an institution for the reproduction and stewardship of human mathematical expertise rather than primarily as a mechanism for producing theorems.

math.HO↗

arXiv2026

Hyperbolicity and obstructions to PL actions of the circle

We investigate acylindrically hyperbolic groups acting on the circle by piecewise linear homeomorphisms. We prove that a finitely generated acylindrically hyperbolic group acting faithfully by piecewise linear homeomorphisms of the circle is virtually free, is virtually a closed hyperbolic surface group, or virtually splits over a two-ended subgroup; as a consequence, we prove a general structural result for Gromov hyperbolic groups of piecewise linear homeomorphisms. Along the way, we show that a right-angled Artin subgroup of piecewise linear homeomorphisms of the circle is either free or abelian, generalizing a result of Bleak and Salazar-Diaz. We also show that the fundamental group of a finite volume hyperbolic $n$--manifold cannot act faithfully on the circle by piecewise linear homeomorphisms, provided that $n\geq 3$, complementing $2$--dimensional examples of Ghys and Minakawa.

math.GR↗

arXiv2026

Cyclotomic polynomials and homological criteria for mapping class types

Let $S_g$ be a closed orientable surface and let $Ψ\colon \mathrm{Mod}(S_g)\to \mathrm{Sp}(2g,\mathbb{Z})$ be the representation induced by the action on first homology. We investigate the characteristic polynomials of integral symplectic matrices arising from mapping classes of algebraically finite type and give a complete characterization in the cyclotomic case: for $n\geq 3$, the polynomial $φ_n(x)$ is realized by a mapping class of algebraically finite type if and only if $n$ has at most two distinct prime divisors. Consequently, if $n$ is square-free and has at least three distinct prime divisors, then every mapping class with characteristic polynomial $φ_n(x)$ is pseudo-Anosov. This gives a cyclotomic complement to the Casson--Bleiler homological criterion and yields a complete criterion for a symplectic polynomial to be realized only by pseudo-Anosov mapping classes.

math.GT↗

arXiv2026

Set theory, logic, and homeomorphism groups of manifolds

We investigate the relationship between axiomatic set theory and the first-order theory of homeomorphism groups of manifolds in the language of group theory, concentrating on first-order rigidity and type versus conjugacy. We prove that under the axiom of constructibility (i.e.~{V=L}), homeomorphism groups of arbitrary connected manifolds are first-order rigid, and that the conjugacy class of a homeomorphism of a manifold is determined by its type. In contradistinction, under the regularity hypothesis that every projective set of reals has the Baire property, we show that in all dimensions greater than one there exist pairs of noncompact, connected manifolds whose homeomorphism groups are elementarily equivalent but which are not homeomorphic. We also show, under the same Baire-property hypothesis, that every manifold of positive dimension admits pairs of homeomorphisms with the same type which are not conjugate to each other. Projective determinacy implies the Baire-property hypothesis, so the corresponding consequences under PD follow immediately. Finally, we show that infinitary formulas do determine conjugacy classes of homeomorphisms and homeomorphism types of manifolds; specifically, the conjugacy class of a homeomorphism of an arbitrary manifold is determined by a single $L_{ω_1ω}$ formula. Similarly, the homeomorphism type of an arbitrary connected manifold is determined by a single $L_{ω_1ω}$ sentence.

math.GT↗

arXiv2026

Right-angled Artin groups of large girth and finite volume hyperbolic $3$--manifold groups

Let $Γ$ be a finite simplicial graph of girth at least five. In this short note, we give a proof that if $M$ is a finite volume hyperbolic $3$--manifold, then the right-angled Artin group $A(Γ)$ cannot contain $π_1(M)$ as a subgroup; the argument is elementary, modulo the resolution of the Virtual Fibering Conjecture and a splitting theorem due to Belegradek. In particular, if $C_n$ denotes the $n$--cycle then $A(C_n)$ cannot contain a finite volume hyperbolic $3$--manifold group for any $n\geq 3$, thus answering a question of A.~Reid.

math.GT↗

arXiv2026

Linearity criteria for automorphism groups of malabelian groups

Let $G$ be a finitely generated malabelian group, let $A\leq\mathrm{Out}(G)$ be a finitely generated subgroup, and let $Γ_{G,A}$ denote the preimage of $A$ in $\mathrm{Aut}(G)$. We give a general criterion for the linearity of $Γ_{G,A}$ in terms of surjections from $G$ to finite simple groups of Lie type.

math.GR↗

arXiv2026

Elementary equivalence and diffeomorphism groups of smooth manifolds

Let $M$ and $N$ be smooth manifolds, with $M$ closed and connected. If the $C^r$--diffeomorphism group of $M$ is elementarily equivalent to the $C^s$--diffeomorphism group of $N$ for some $r,s\in[1,\infty)\cup\{0,\infty\}$, then $r=s$ and $M$ and $N$ are $C^r$--diffeomorphic. This strengthens a previously known result by Takens and Filipkiewicz, which asserts that for integer regularities, a group isomorphism between diffeomorphism groups of closed manifolds necessarily arises from a diffeomorphism of the underlying manifolds. We prove an analogous result for groups of diffeomorphisms preserving smooth volume forms, in dimension at least two.

math.GR↗

arXiv2025

Generic torsion-free groups and Rubin actions

We use model theoretic forcing to prove that a generic countable torsion-free group does not admit any nontrivial locally moving action on a Hausdorff topological space, and yet admits a rich Rubin poset.

math.GR↗

arXiv2025

Uniform first order interpretation of the second order theory of countable groups of homeomorphisms

We show that the first order theory of the homeomorphism group of a compact manifold interprets the full second order theory of countable groups of homeomorphisms of the manifold. The interpretation is uniform across manifolds of bounded dimension. As a consequence, many classical problems in group theory and geometry (e.g.~the linearity of mapping classes of compact $2$--manifolds) are encoded as elementary properties of homeomorphism groups of manifolds. Furthermore, the homeomorphism group uniformly interprets the Borel and projective hierarchies of the homeomorphism group, which gives a characterization of definable subsets of the homeomorphism group. Finally, we prove analogues of Rice's Theorem from computability theory for homeomorphism groups of manifolds. As a consequence, it follows that the collection of sentences that isolate the homeomorphism group of a particular manifold, or that isolate the homeomorphism groups of manifolds in general, is not definable in second order arithmetic, and that membership of particular sentences in these collections cannot be proved in ZFC.

math.GR↗

arXiv2025

First order rigidity of homeomorphism groups of manifolds

For every compact, connected manifold $M$, we prove the existence of a sentence $ϕ_M$ in the language of groups such that the homeomorphism group of another compact manifold $N$ satisfies $ϕ_M$ if and only if $N$ is homeomorphic to $M$. We prove the analogous statement for groups of homeomorphisms preserving an Oxtoby--Ulam probability measure.

math.GR↗

arXiv2024

Locally approximating groups of homeomorphisms of manifolds

Let $M$ be a compact, connected manifold of positive dimension and let $\mathcal G\leq\textrm{Homeo}(M)$ be \emph{locally approximating} in the sense that for all open $U\subseteq M$ compactly contained in a single Euclidean chart of $M$, the subgroup $\mathcal G[U]$ consisting of elements of $\mathcal G$ supported in $U$ is dense in the full group of homeomorphisms supported in $U$. We prove that $\mathcal G$ interprets first order arithmetic, as well as a first order predicate that encodes membership in finitely generated subgroups of $\mathcal G$. As a consequence, we show that if $\mathcal G$ is not finitely generated, then no group elementarily equivalent to $\mathcal G$ can be finitely generated. We show that many finitely generated locally approximating groups of homeomorphisms $\mathcal G$ of a manifold are prime models of their theories, and give conditions that guarantee any finitely presented group $G$ that is elementarily equivalent to $\mathcal G$ is isomorphic to $\mathcal G$. We thus recover some results of Lasserre about the model theory of Thompson's groups $F$ and $T$. Finally, we obtain several action rigidity result for locally approximating groups of homeomorphisms. If $\mathcal G$ acts in a locally approximating way on a compact, connected manifold $M$ then the dimension of $M$ is uniquely determined by the elementary equivalence class of $\mathcal G$. Moreover, if $\dim M\leq 3$ then $M$ is uniquely determined up to homeomorphism. In for general closed smooth manifolds, the homotopy type of $M$ is uniquely determined. In this way, we obtain a generalization of a well-known result of Rubin.

math.GR↗

arXiv2024

Post-quantum hash functions using $\mathrm{SL}_n(\mathbb{F}_p)$

We define new families of Tillich-Zémor hash functions, using higher dimensional special linear groups over finite fields as platforms. The Cayley graphs of these groups combine fast mixing properties and high girth, which together give rise to good preimage and collision resistance of the corresponding hash functions. We justify the claim that the resulting hash functions are post-quantum secure.

cs.CR↗

arXiv2024

Right-angled Artin groups and the cohomology basis graph

Let $Γ$ be a finite graph and let $A(Γ)$ be the corresponding right-angled Artin group. From an arbitrary basis $\mathcal B$ of $H^1(A(Γ),\mathbb F)$ over an arbitrary field, we construct a natural graph $Γ_{\mathcal B}$ from the cup product, called the \emph{cohomology basis graph}. We show that $Γ_{\mathcal B}$ always contains $Γ$ as a subgraph. This provides an effective way to reconstruct the defining graph $Γ$ from the cohomology of $A(Γ)$, to characterize the planarity of the defining graph from the algebra of $A(Γ)$, and to recover many other natural graph-theoretic invariants. We also investigate the behavior of the cohomology basis graph under passage to elementary subminors, and show that it is not well-behaved under edge contraction.

math.GR↗

arXiv2023

The model theory of the curve graph

In this paper we develop a bridge between model theory, geometric topology, and geometric group theory. In particular, we investigate the Ivanov Metaconjecture from the point of view of model theory, and more broadly we seek to answer the general question: why does the curve graph of a surface play such a central role in the study of surfaces and mapping class groups? More specifically, we consider a surface $Σ$ of finite type and its curve graph $\mathcal C(Σ)$, and we investigate its first-order theory in the language of graph theory. Crucially, $\mathcal C(Σ)$ is bi-interpretable with a certain object called the augmented Cayley graph of the mapping class group of the surface. We use this bi-interpretation to prove that the theory of the curve graph is $ω$--stable, to compute its Morley rank, and to show that it has quantifier elimination with respect to the class of $\forall\exists$--formulae. We also show that many of the complexes which are naturally associated to a surface are interpretable in $\mathcal C(Σ)$. This shows that these complexes are all $ω$--stable and admit certain a priori bounds on their Morley ranks. We are able to use Morley ranks to prove that various complexes are not bi--interpretable with the curve graph. As a consequence of quantifier elimination, we show that algebraic intersection number is not definable in the first order theory of the curve graph. Finally, we prove that the curve graph of a surface enjoys a novel phenomenon that we call interpretation rigidity. That is, if surfaces $Σ_1$ and $Σ_2$ admits curve graphs that are mutually interpretable, then $Σ_1$ and $Σ_2$ are homeomorphic to each other. Along the way, numerous technical results are obtained.

math.GT↗

arXiv2023

Commensurators of thin normal subgroups and abelian quotients

We give an affirmative answer to many cases of a question due to Shalom, which asks if the commensurator of a thin subgroup of a Lie group is discrete. In this paper, let $K<Γ<G$ be an infinite normal subgroup of an arithmetic lattice $Γ$ in a rank one simple Lie group $G$, such that the quotient $Q=Γ/K$ is infinite. We show that the commensurator of $K$ in $G$ is discrete, provided that $Q$ admits a surjective homomorphism to $\mathbb{Z}$. In this case, we also show that the commensurator of $K$ contains the normalizer of $K$ with finite index. We thus vastly generalize a result of the authors, which showed that many natural normal subgroups of $\mathrm{PSL}_2(\mathbb{Z})$ have discrete commensurator in $\mathrm{PSL}_2(\mathbb{R})$.

math.GT↗

arXiv2022

Virtual critical regularity of mapping class group actions on the circle

We show that if $G_1$ and $G_2$ are non-solvable groups, then no $C^{1,τ}$ action of $(G_1\times G_2)*\mathbb{Z}$ on $S^1$ is faithful for $τ>0$. As a corollary, if $S$ is an orientable surface of complexity at least three then the critical regularity of an arbitrary finite index subgroup of the mapping class group $\mathrm{Mod}(S)$ with respect to the circle is at most one, thus strengthening a result of the first two authors with Baik.

math.GR↗

arXiv2021

Expanders and right-angled Artin groups

The purpose of this article is to give a characterization of families of expander graphs via right-angled Artin groups. We prove that a sequence of simplicial graphs $\{Γ_i\}_{i\in\mathbb{N}}$ forms a family of expander graphs if and only if a certain natural mini-max invariant arising from the cup product in the cohomology rings of the groups $\{A(Γ_i)\}_{i\in\mathbb{N}}$ agrees with the Cheeger constant of the sequence of graphs, thus allowing us to characterize expander graphs via cohomology. This result is proved in the more general framework of \emph{vector space expanders}, a novel structure consisting of sequences of vector spaces equipped with vector-space-valued bilinear pairings which satisfy a certain mini-max condition. These objects can be considered to be analogues of expander graphs in the realm of linear algebra, with a dictionary being given by the cup product in cohomology, and in this context represent a different approach to expanders that those developed by Lubotzky-Zelmanov and Bourgain-Yehudayoff.

math.GR↗

arXiv2021

Hamiltonicity via cohomology of right-angled Artin groups

Let $Γ$ be a finite graph and let $A(Γ)$ be the corresponding right-angled Artin group. We characterize the Hamiltonicity of $Γ$ via the structure of the cohomology algebra of $A(Γ)$. In doing so, we define and develop a new canonical graph associated to a matrix, which as a consequence provides a novel perspective on the matrix determinant.

math.GR↗