Papers

Let $G$ be a finite group admitting a rational fixed-point-free (FPF) representation $V$, with a $G$-invariant full-rank lattice $L\subset V$. We study the induced $G$-action on the complex torus $V_{\mathbb{C}}/L$ and the quotient orbifold $V_{\mathbb{C}}/(G\ltimes L)$. We prove that every FPF group admits a unique irreducible rational FPF representation. We derive closed-form formulas for three core geometric invariants: the number of non-free points of the torus action, the number of singular points of the quotient orbifold, and its orbifold Euler characteristic.

We establish the extension of Milnor's inequality, relating total curvature with the bridge index, to the $C^1$-closure of a knot class, without any restriction on the self-intersections of the limit curve. Together with recent work of Reiter–von der Mosel, this resolves the circular elastic knot conjecture, first predicted by Gallotti–Pierre-Louis in 2007 and then formulated as a mathematical conjecture by Gerlach–Reiter–von der Mosel in 2017. More precisely, if the bridge and braid indices of a tame knot class coincide, then the multiply covered circle is the unique elastic knot.

We resolve the longstanding geometric ideal triangulation conjecture arising from Thurston's theory of hyperbolic 3-manifolds: every finite-volume cusped hyperbolic 3-manifold admits a finite geometric ideal triangulation realizing its complete hyperbolic structure.

To each modular knot, we attach a weight $2$ modular integral for $\mathrm{SL}_2(\mathbb{Z})$ whose homogenized cycle integrals recover its linking numbers with other modular knots in $S^3$. This construction answers Choie-Zagier's question of explicitly constructing modular integrals with prescribed rational period functions in weight $2$. Our formula applies to individual modular knots without symmetrization, thereby realizing an approach discussed but not pursued by Duke-Imamoḡlu-Tóth. This gives an answer to Ghys' question on pairwise linking numbers in terms of modular integrals, as sought by Simon.

In this note we develop a novel approach for proving the non-vanishing of bounded cohomology. This utilizes a splitting argument whose simplest form is as follows: Let M denote an n-manifold of non-zero simplicial volume and N a codimension two submanifold of M, then one can conclude that the n-th bounded cohomology of the fundamental group of M \ N is non-zero. We then translate the existence of a complement with a given fundamental group into an easily accessible homology computation, which might be of independent interest.

We show the smallest non-abelian quotients of spherical and affine Artin groups are isomorphic to the smallest non-abelian quotients of the corresponding Coxeter groups. We deduce irreducible spherical Artin groups are determined by their finite quotient groups.

For type $A$ braid varieties, including Richardson varieties and positroid varieties, we prove that the standard torus has a dense orbit exactly when the associated link is smoothly slice. More generally, we relate the smooth slice genus to the codimension of a generic standard-torus orbit. For knots, this codimension equals twice the slice genus. We also characterize algebraic tori among open affine Richardson varieties and positroid patches by 2-crown avoidance in their Bruhat intervals. Finally, positroid links have equal Seifert and slice genera, so a positroid link is smoothly slice exactly when it is an unlink.

Let $\Gamma<\mathsf{SL}(d,\mathbb R)$ be a Zariski-dense Borel-Anosov subgroup and let $\varphi$ be positive on its limit cone. For every $\chi \in \mathrm{Hom}(\Gamma, \mathbb R)$, we construct twisted $(\varphi,\chi)$-conformal densities and prove their uniqueness, atomlessness, and ergodicity. For every normal subgroup $\Gamma_0\lhd\Gamma$ with nilpotent quotient, we classify the ergodic $\varphi$-conformal densities of $\Gamma_0$ in terms of characters of $\Gamma/\Gamma_0$.

We prove a sharp effective equidistribution theorem for closed geodesics on a compact hyperbolic surface. This is achieved by carefully analyzing the Selberg trace formula twisted by a Maass eigenfunction, which we refer to as Zelditch's trace formula, following Zelditch's observation that the geometric side can be expressed as a sum of period integrals of eigenfunctions on closed geodesics. The main technical input is a sharp upper bound for the matrix coefficients $\langle \phi, \varphi^2 \rangle$ uniformly in $t_\phi\to \infty$ and $t_\varphi \to \infty$, where $t_\phi$ and $t_\varphi$ are the eigenparameters corresponding to Maass forms $\phi$ and $\varphi$ on $X$.

We give a complete classification of algebraically primitive Teichmüller curves in $\Omega\mathcal M_g(2g-2)^{\mathrm{hyp}}$ for $g\ge5$. They are precisely the curves generated by the double regular \((2g+1)\)-gon when \(2g+1\) is prime, and by the regular \(4g\)-gon when \(g\) is a power of two.

We present a new algorithm to compute a stream for the length spectrum of an oriented, finite-volume hyperbolic 3-manifold $M$. The advantage over the existing algorithm by Hodgson-Weeks is that our algorithm does not require finding a Dirichlet domain and thus can use interval arithmetic to verify the result. Our algorithm is also significantly faster in some cases. We also present a new algorithm to compute the optimal Margulis number of $M$. The authors have made these algorithms available in SnapPy.

Suppose that a discrete group $G$ acts smoothly, properly, and effectively on a connected smooth manifold $M$, and let $q:M\to M/G$ be the quotient map. We prove that if $q$ admits a continuous section, then $G$ is generated by reflections. We also establish converse results for dissecting reflections and give applications to Euclidean and lattice actions.

We construct infinitely many pairwise non-diffeomorphic irreducible smooth four-manifolds homeomorphic to ${\mathbb {CP}}^2\# 7{\overline {\mathbb {C}} {\mathbb {P}} } ^2$, each admitting a free, orientation-preserving involution. Their quotients give infinitely many irreducible definite four-manifolds with fundamental group ${\mathbb {Z}}/2 {\mathbb {Z}}$ and $b_2=3$, filling the corresponding gap in the geography of such examples. The construction of infinitely many distinct examples relies on a novel equivariant torus surgery.

Let $\Sigma$ be a compact oriented surface. We investigate the graded Lie algebra of homology cylinders over $\Sigma$, as introduced by M. Goussarov and K. Habiro. To analyze its torsion, we refine a strategy initiated by Y. Nozaki, M. Suzuki, and the third author, which relies on the reduction modulo $1$ of the LMO functor and on clasper calculus. Specifically, we develop general tools for studying, under the standard action of the symplectic group, the Sp-module structure of the torsion in the odd-degree component of this Lie algebra. We show that this torsion part surjects onto an $2$-torsion Sp-module, which is explicitly described in terms of Jacobi diagrams. As an application, we provide an intrinsic description of the Sp-module given by the degree-three component of the Lie algebra of homology cylinders. A further motivation is to understand torsion phenomena in the associated graded of the lower central series of the Torelli group of $\Sigma$: in this direction, we exhibit an explicit Sp-module onto which the torsion of the Torelli Lie algebra surjects in degree three.

We prove that irreducible spherical type Artin groups are profinitely rigid within the class of all spherical type Artin groups. As part of the proof, building on ideas of Kolay for the $n$-strand braid group, we compute the smallest non-abelian quotient of every spherical type Artin group. In particular, given an irreducible spherical type Artin group, its smallest non-abelian quotient is always that of the corresponding Coxeter group, except for dihedral Artin groups of weight divisible by $4$, which all have $\mathfrak S_3$ as a non-abelian quotient. Parts of the proof also rely on the computation of the cohomological dimension of the profinite completion of certain Artin groups.

Generalised $T$-links provide a simple description of all links in $S^3$ as closures of products of standard twisting blocks, parametrised by finite sequences of integers. We prove that every link admits infinitely many pairwise distinct generalised $T$-link presentations. Starting from any such presentation, we give explicit parameter transformations that preserve the represented link and generate families of pairwise distinct presentations depending on arbitrarily many independent integer parameters.

We prove the following: If $X$ is a plane separating continuum, then every embedding of $X$ into $\mathbb{R}^2$ contains the vertices of a Euclidean rectangle. We arrive to this result by extending a known result by H. Vaughan for Jordan curves to a wider class of topological objects via shape theory and Steenrod homology.

The is an expanded version of a memorial article that is to appear in the Notices of the American Mathematical Society.

We prove the existence of weight 2 modular forms for some Veech groups, whose Fourier coefficients at a cusp determine the counts of numbers of branched Galois covers of punctured elliptic curves. We further prove there exist cusp forms, counting ramified genus $2$ covers, that are not invariant under any congruence subgroup of SL(2,Z). The main results may be viewed as analogs of the construction of an elliptic curve obtained as the quotient of the Jacobian of a classical modular curve, whose point counts modulo primes determine the coefficients of a Hecke eigenform, but where the uniformizing groups may be noncongruence.

We study a logarithmic re-gauging of the visual boundary, arising from the logarithmic metric on the sublinear Morse boundary introduced by Garg-Jana-Qing. For every proper geodesic hyperbolic space $X$, we prove that $d_{\mathrm{log}}(\xi,\eta)\asymp (1+(\xi\mid\eta)_o)^{-\theta}$. As a consequence, under the natural identification of boundaries, $d_{\mathrm{log}}$ is bi-Lipschitz equivalent to the polynomial Floyd boundary metric associated with the radial density $g_\theta(t)=(1+t)^{-1-\theta}$. Every quasi-isometry between proper geodesic hyperbolic spaces induces a bi-Lipschitz map between their logarithmic boundaries with a fixed admissible exponent. For every non-elementary hyperbolic group, the logarithmic boundary is nondoubling and has infinite Hausdorff and Assouad dimensions. Nevertheless, the logarithmic re-gauging preserves Nagata (capacity) dimension and continues to recover the asymptotic dimension of the group. It also detects one-endedness through linear connectedness (bounded turning) and preserves uniform perfectness. Thus the logarithmic and visual metrics are not quasisymmetrically equivalent, although several coarse-geometric features remain visible in the logarithmic boundary.

For every infinite-type surface with an isolated puncture that admits a shift map and every natural number $n$, we study an intrinsically infinite-type mapping class $g_n$ and a simple closed curve whose orbit under $g_n$ splits into $n$ subsequences, each of which converges in the coarse Chabauty topology to a geodesic lamination on the surface. These laminations are cyclically permuted by $g_n$, and their union is a $g_n$–invariant geodesic lamination. We further show the attracting limit point of $g_n$ in the boundary of the relative arc graph has weight $n$. In particular, intrinsically infinite-type loxodromic isometries of the relative arc graph realize every finite weight.

We give a short proof of the following extension of a recent result of Gadre, Maher, Pfaff and Uyanki: Consider a closed hyperbolic 3-manifold M which fibers over the circle, with fiber a closed surface S. Any measure class on the ideal boundary of hyperbolic 3-space which is invariant and totally ergodic under the action of the fundamental group of M is singular with respect to the push-forward by a Cannon-Thurston map of a stationary measure on the circle of a random walk with finite first moment on the fundamental group of S.

We introduce collision rank and show that groups of finite collision rank are good in the sense of topological $4$-manifold theory. Finitely generated groups of collision rank two are shown to exist outside the class of subexponentially amenable groups. Such examples are given by commutator subgroups of topological full groups of Sturmian systems, and include continuum many pairwise nonisomorphic infinite simple amenable groups.

We define product subcomplexes and intersection complexes for compact nonpositively curved square complexes and CAT(0) square complexes. Product subcomplexes are defined as equivalence classes of local isometries from products of graphs with embedded coordinate fibers. Using their factorizations, we give a common definition of the intersection complex, extending the constructions in S. Oh's paper "Quasi-isometry invariants of weakly special square complexes" (Topology and its Applications 307 (2022)). Cubical automorphisms induce actions on these complexes. We define morphisms by comparing compatible elevations of core-map labels and their factorizations along face chains. For a compact two-sided weakly special square complex, we prove that its intersection complex is canonically isomorphic to the deck quotient of the intersection complex of its universal cover. Under a simplicity hypothesis, this recovers the image-defined reduced intersection complex in that paper, and product-base groups are the full simplex stabilizers. We also prove that quasi-isometries between universal covers of compact two-sided weakly special square complexes induce isomorphisms of their intersection complexes in this sense, and give finite counterexamples explaining why images alone do not determine the quotient in general.

We study fixed point theorems for actions of lattices of semisimple groups. They are deduced from vanishing results for the group cohomology of $L^p$-representations. We show that for lattices in simple groups of higher rank, the cohomology with $L^p$-coefficients vanishes below the rank whenever there are no invariant vectors. As a corollary of the vanishing for $L^1$-coefficients, we obtain that every action on an acyclic simplicial complex of dimension lower than the rank has a finite orbit. This in particular proves a conjecture by Farb. The $L^p$-vanishing below the rank proves a conjecture by Gromov regarding $L^p$-cohomology of symmetric spaces.

We determine the exact probability that the closed hexagonal polygon obtained by cyclically joining six independent points uniformly distributed on the unit sphere is knotted. The only possible nontrivial knot is a trefoil. Almost surely, the convex hull of the six points has one of two simplicial combinatorial types: a combinatorially regular octahedron and a combinatorially non-regular octahedron. We show that the straight-line complete graph on the six vertices of the hull of the combinatorially regular octahedral type admits exactly one unoriented trefoil Hamiltonian cycle. On the other hand, no cycle connecting vertices of the irregular octahedron can produce a trefoil. We then use stereographic projection to transform the probability of the regular hull type to a Sylvester-type problem for the planar beta-prime probability measure $d\mu(x,y)=\frac{dx dy}{\pi\ (1+x^2+y^2)^2}$. Applying Stokes' theorem and the Blaschke-Petkantschin formula, we compute the expected squared $\mu$-content of a random triangle. This yields a regular octahedral probability of $\frac{15}{4\pi^2}$ and a trefoil probability of $\frac{1}{16\pi^2}$.

In this paper, we find infinitely many examples of nonsmoothable embedded surfaces in a 4-manifold that become smoothable after taking the connected sum with S^2 x S^2. To the author's knowledge, no such examples have previously been discovered. In [CK25], Cha and Kim asked for the minimal number of stabilizations required to turn a nonsmoothable surface into a smoothable one. We provide sufficient conditions ensuring that this number is 1. In addition, we show that there is a nonsmoothable surface in a K3 surface such that, after a single stabilization, its topological isotopy class contains infinitely many smooth representatives that are pairwise not smoothly isotopic.

Watanabe constructed many nontrivial family diffeomorphisms in $\pi_*\text{Diff}_\partial(D^4)$, which were obtained by doing clasper surgeries on trivalent graphs. In this paper, we generalize the barbell diffeomorphism discovered by Budney and Gabai and construct a series of nontrivial elements in $\pi_{k-1}\text{Diff}_\partial(M_{k+1}')$ with $M_{k+1}'=\natural_{k+1} S^2\times D^2$ the $(k+1)$-cuff barbell, which we call higher barbell diffeomorphisms. We show that Watanabe's constructions can be realized as implanted higher barbell diffeomorphisms in $D^4$ if the trivalent graph is homologically nonzero in the graph complex. Combining this realization with Watanabe's detection theorem, we establish the nontriviality of these implanted higher barbell diffeomorphisms.

For every admissible non-maximal Toledo invariant $t$, we construct a locus of irreducible representations of a closed genus-$g$ surface group into $\mathrm{PU}_{2,n+1}$ with Toledo invariant $t$ whose orbit maps are quasi-isometric embeddings. These loci have real codimension $10(g-1)$ or $10(g-1)+2(n-1)$ in the character variety depending on the Toledo value. Our construction uses $4$-cyclic Higgs bundles and their correspondence with $\partial$-alternating surfaces in complex pseudo-hyperbolic spaces. The associated equivariant minimal maps into the symmetric space are bi-Lipschitz embeddings. We further construct cocompact domains of discontinuity in the Shilov boundary. Their quotients are smooth fiber bundles over the surface with fiber homeomorphic to $S^{2n-1}\times S^{2n-1}$.

We give a complete classification of algebraically primitive Teichmüller curves in $\Omega\mathcal M_g(g-1,g-1)^{\mathrm{hyp}}$ for $g>2$. These curves are precisely those generated by the Veech $2p$-gons, where $p=2g+1$ is prime.