arXiv subjects
Publications and source records attributed to Thomas Koberda.
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.
In this monograph, we give an account of the relationship between the algebraic structure of finitely generated and countable groups and the regularity with which they act on manifolds. We concentrate on the case of one--dimensional manifolds, culminating with a uniform construction of finitely generated groups acting with prescribed regularity on the compact interval and on the circle. We develop the theory of dynamical obstructions to smoothness, beginning with classical results of Denjoy, to more recent results of Kopell, and to modern results such as the $abt$--Lemma. We give a classification of the right-angled Artin groups that have finite critical regularity and discuss their exact critical regularities in many cases, and we compute the virtual critical regularity of most mapping class groups of orientable surfaces.
Let $H<\mathrm{PSL}_2(\mathbb{Z})$ be a finite index normal subgroup which is contained in a principal congruence subgroup, and let $Φ(H)\neq H$ denote a term of the lower central series or the derived series of $H$. In this paper, we prove that the commensurator of $Φ(H)$ in $\mathrm{PSL}_2(\mathbb{R})$ is discrete. We thus obtain a natural family of thin subgroups of $\mathrm{PSL}_2(\mathbb{R})$ whose commensurators are discrete, establishing some cases of a conjecture of Shalom.
We survey the relationship between the combinatorics and geometry of graphs and the algebraic structure of right-angled Artin groups. We concentrate on the defining graph of the right-angled Artin group and on the extension graph associated to the right-angled Artin group. Additionally, we discuss connections to geometric group theory and complexity theory. The final version of this survey will appear in "In the tradition of Thurston, vol.~II", ed.~K.~Ohshika and A.~Papadopoulos.
We address the problem of computing the critical regularity of groups of homeomorphisms of the interval. Our main result is that if $H$ and $K$ are two non-solvable groups then a faithful $C^{1,τ}$ action of $H\times K$ on a compact interval $I$ is {\em not overlapping} for all $τ>0$, which by definition means that there must be non-trivial $h\in H$ and $k\in K$ with disjoint support. As a corollary we prove that the right-angled Artin group $(F_2\times F_2)*\mathbb{Z}$ has critical regularity one, which is to say that it admits a faithful $C^1$ action on $I$, but no faithful $C^{1,τ}$ action. This is the first explicit example of a group of exponential growth which is without nonabelian subexponential growth subgroups, whose critical regularity is finite, achieved, and known exactly. Another corollary we get is that Thompson's group $F$ does not admit a faithful $C^1$ overlapping action on $I$, so that $F*\mathbb{Z}$ is a new example of a locally indicable group admitting no faithful $C^1$--action on $I$.
Let $Δ$ be a hyperbolic triangle with a fixed area $φ$. We prove that for all but countably many $φ$, generic choices of $Δ$ have the property that the group generated by the $π$--rotations about the midpoints of the sides of the triangle admits no nontrivial relations. By contrast, we show for all $φ\in(0,π)\setminus\mathbb{Q}π$, a dense set of triangles does afford nontrivial relations, which in the generic case map to hyperbolic translations. To establish this fact, we study the deformation space $\mathfrak{C}_θ$ of singular hyperbolic metrics on a torus with a single cone point of angle $θ=2(π-φ)$, and answer an analogous question for the holonomy map $ρ_ξ$ of such a hyperbolic structure $ξ$. In an appendix by X.~Gao, concrete examples of $θ$ and $ξ\in\mathfrak{C}_θ$ are given where the image of each $ρ_ξ$ is finitely presented, non-free and torsion-free; in fact, those images will be isomorphic to the fundamental groups of closed hyperbolic 3--manifolds.
We characterize $k$--colorability of a simplicial graph via the intrinsic algebraic structure of the associated right-angled Artin group. As a consequence, we show that a certain problem about the existence of homomorphisms from right-angled Artin groups to products of free groups is NP--complete.
We study the regularity of exceptional actions of groups by $C^{1,α}$ diffeomorphisms on the circle, i.e. ones which admit exceptional minimal sets, and whose elements have first derivatives that are continuous with concave modulus of continuity $α$. Let $G$ be a finitely generated group admitting a $C^{1,α}$ action $ρ$ with a free orbit on the circle, and such that the logarithms of derivatives of group elements are uniformly bounded at some point of the circle. We prove that if $G$ has spherical growth bounded by $c n^{d-1}$ and if the function $1/α^d$ is integrable near zero, then under some mild technical assumptions on $α$, there is a sequence of exceptional $C^{1,α}$ actions of $G$ which converge to $ρ$ in the $C^1$ topology. As a consequence for a single diffeomorphism, we obtain that if the function $1/α$ is integrable near zero, then there exists a $C^{1,α}$ exceptional diffeomorphism of the circle. This corollary accounts for all previously known moduli of continuity for derivatives of exceptional diffeomorphisms. We also obtain a partial converse to our main result. For finitely generated free abelian groups, the existence of an exceptional action, together with some natural hypotheses on the derivatives of group elements, puts integrability restrictions on the modulus $α$. These results are related to a long-standing question of D. McDuff concerning the length spectrum of exceptional $C^1$ diffeomorphisms of the circle.
Let $q\in\mathbb{C}$, let \[a=\begin{pmatrix} 1&0\\1&1\end{pmatrix},\quad b_q=\begin{pmatrix} 1&q\\0&1\end{pmatrix},\] and let $G_q<\mathrm{SL}_2(\mathbb{C})$ be the group generated by $a$ and $b_q$. In this paper, we study the problem of determining when the group $G_q$ is not free for $|q|<4$ rational. We give a robust computational criterion which allows us to prove that if $q=s/r$ for $|s|\leq 27$ then $G_q$ is non-free, with the possible exception of $s=24$. In this latter case, we prove that the set of denominators $r\in\mathbb{N}$ for which $G_{24/r}$ is non-free has natural density $1$. For a general numerator $s>27$, we prove that the lower density of denominators $r\in \mathbb{N}$ for which $G_{s/r}$ is non-free has a lower bound \[ 1- \left(1-\frac{11}{s}\right) \prod_{n=1}^\infty \left(1-\frac{4}{s^{2^n-1}}\right). \] Finally, we show that for a fixed $s$, there are arbitrarily long sequences of consecutive denominators $r$ such that $G_{s/r}$ is non-free. The proofs of some of the results are computer assisted, and Mathematica code has been provided together with suitable documentation.
Let $T$ be a compact fibered $3$--manifold, presented as a mapping torus of a compact, orientable surface $S$ with monodromy $ψ$, and let $M$ be a compact Riemannian manifold. Our main result is that if the induced action $ψ^*$ on $H^1(S,\mathbb{R})$ has no eigenvalues on the unit circle, then there exists a neighborhood $\mathcal U$ of the trivial action in the space of $C^1$ actions of $π_1(T)$ on $M$ such that any action in $\mathcal{U}$ is abelian. We will prove that the same result holds in the generality of an infinite cyclic extension of an arbitrary finitely generated group $H$, provided that the conjugation action of the cyclic group on $H^1(H,\mathbb{R})\neq 0$ has no eigenvalues of modulus one. We thus generalize a result of A. McCarthy, which addressed the case of abelian--by--cyclic groups acting on compact manifolds.
Let $M$ be the circle or a compact interval, and let $α=k+τ\ge1$ be a real number such that $k=\lfloor α\rfloor$. We write $\mathrm{Diff}_+^α(M)$ for the group of $C^k$ diffeomorphisms of $M$ whose $k^{th}$ derivatives are Hölder continuous with exponent $τ$. If $α\ge1$, we prove that there exists a finitely generated subgroup $G_α\le\mathrm{Diff}_+^α(M)$ with the property that $G_α$ admits no injective homomorphisms into $\mathrm{Diff}_+^β(M)$ for all $β>α$. If $α>1$, we also show the dual result: there exists a finitely generated group $H_α\le\bigcap_{β<α}\mathrm{Diff}_+^β(M)$ with the property that $H_α$ admits no injective homomorphisms into $\mathrm{Diff}_+^α(M)$. We can further require that the same properties are inherited by all finite index subgroups, and also by the commutator subgroups, of $G_α$ and $H_α$. The commutator groups of $G_α$ and of $H_α$ are countable simple groups. As a consequence, whenever $1\leα<β$ we have a continuum of isomorphism types of finitely generated subgroups of $\mathrm{Diff}_+^α(M)$ whose images under arbitrary homomorphisms to $\mathrm{Diff}_+^β(M)$ are abelian. We give some applications to smoothability of codimension one foliations and to homomorphisms between certain continuous groups of diffeomorphisms. For example, we show that if $k\neq 2$ is an integer and if $k<β$ then there is no nontrivial homomorphism $\mathrm{Diff}_+^k(S^1)\to\mathrm{Diff}_+^β(S^1)$.
We discuss some applications of 3-manifold topology to cryptography. In particular, we propose a public-key and a symmetric-key cryptographic scheme based on the Thurston norm on the first cohomology of hyperbolic manifolds.
We establish an effective version of the classical Lie--Kolchin Theorem. Namely, let $A,B\in\mathrm{GL}_m(\mathbb{C})$ be quasi--unipotent matrices such that the Jordan Canonical Form of $B$ consists of a single block, and suppose that for all $k\geq0$ the matrix $AB^k$ is also quasi--unipotent. Then $A$ and $B$ have a common eigenvector. In particular, $\langle A,B\rangle<\mathrm{GL}_m(\mathbb{C})$ is a solvable subgroup. We give applications of this result to the representation theory of mapping class groups of orientable surfaces.
It is a celebrated result of Mather that the group of $C^k$--diffeomorphisms of an $n$--manifold is simple, provided that a mild isotopy condition is satisfied, with the possible exception of $k=n+1$. The purpose of this article is mostly expository, and in it we give a detailed account of Mather's proof in the case when $n=1$.
In this article we develop the theory of residually finite rationally $p$ (RFR$p$) groups, where $p$ is a prime. We first prove a series of results about the structure of finitely generated RFR$p$ groups (either for a single prime $p$, or for infinitely many primes), including torsion-freeness, a Tits alternative, and a restriction on the BNS invariant. Furthermore, we show that many groups which occur naturally in group theory, algebraic geometry, and in $3$-manifold topology enjoy this residual property. We then prove a combination theorem for RFR$p$ groups, which we use to study the boundary manifolds of algebraic curves $\mathbb{CP}^2$ and in $\mathbb{C}^2$. We show that boundary manifolds of a large class of curves in $\mathbb{C}^2$ (which includes all line arrangements) have RFR$p$ fundamental groups, whereas boundary manifolds of curves in $\mathbb{CP}^2$ may fail to do so.
We study two--generated subgroups $\langle f,g\rangle<\mathrm{Homeo}^+(I)$ such that $\langle f^2,g^2\rangle$ is isomorphic to Thompson's group $F$, and such that the supports of $f$ and $g$ form a chain of two intervals. We show that this class contains uncountably many isomorphism types. These include examples with nonabelian free subgroups, examples which do not admit faithful actions by $C^2$ diffeomorphisms on $1$--manifolds, examples which do not admit faithful actions by $PL$ homeomorphisms on an interval, and examples which are not finitely presented. We thus answer questions due to M. Brin. We also show that many relatively uncomplicated groups of homeomorphisms can have very complicated square roots, thus establishing the behavior of square roots of $F$ as part of a general phenomenon among subgroups of $\mathrm{Homeo}^+(I)$.
In this paper we consider several classical and novel algorithmic problems for right-angled Artin groups, some of which are closely related to graph theoretic problems, and study their computational complexity. We study these problems with a view towards applications to cryptography.
In this partly expository monograph we develop a general framework for producing uncountable families of exotic actions of certain classically studied groups acting on the circle. We show that if $L$ is a nontrivial limit group then the nonlinear representation variety $\mathrm{Hom}(L,\mathrm{Homeo}_+(S^1))$ contains uncountably many semi-conjugacy classes of faithful actions on $S^1$ with pairwise disjoint rotation spectra (except for $0$) such that each representation lifts to $\mathbb{R}$. For the case of most Fuchsian groups $L$, we prove further that this flexibility phenomenon occurs even locally, thus complementing a result of K. Mann. We prove that each non-elementary free or surface group admits an action on $S^1$ that is never semi-conjugate to any action that factors through a finite--dimensional connected Lie subgroup in $\mathrm{Homeo}_+(S^1)$. It is exhibited that the mapping class groups of bounded surfaces have non-semi-conjugate faithful actions on $S^1$. In the process of establishing these results, we prove general combination theorems for indiscrete subgroups of $\mathrm{PSL}_2(\mathbb{R})$ which apply to most Fuchsian groups and to all limit groups. We also show a Topological Baumslag Lemma, and general combination theorems for representations into Baire topological groups. The abundance of $\mathbb{Z}$--valued subadditive defect--one quasimorphisms on these groups would follow as a corollary. We also give a mostly self-contained reconciliation of the various notions of semi-conjugacy in the extant literature by showing that they are all equivalent.