Papers

Habiro's cyclotomic expansion of the colored Jones polynomial has a higher rank analogue conjectured by Chen–Liu–Zhu for colored $SU(n)$ invariants. We prove this conjecture for torus knots $T(2,2p+1)$ and give a Bailey theoretic realization of the cyclotomic coefficients. For $n\geq2$ and $K_p=T(2,2p+1)$, the colored $SU(n)$ invariants admit an expansion $$ J_N^{SU(n)}(K_p;q) = \sum_{m=0}^{N} \left(\prod_{j=0}^{m-1}\{N-j\}\{N+n+j\}\right) H_m^{(n,p)}(q), $$ where $H_m^{(n,p)}(q)\in\mathbb Z[q^{\pm1}]$ is independent of the color $N$. The proof identifies the cyclotomic basis with a Newton basis and rewrites the resulting Newton coefficients as an ordinary Bailey transform. The Lin–Zheng formula for $T(2,2p+1)$ then gives a well-poised Bailey kernel. A terminating very-well-poised ${}_6\phi_5$ summation diagonalizes the kernel and reduces the integrality problem to an ordinary Bailey transition. We prove a uniform integrality theorem for these transitions in a formal integral $q$-difference operator algebra. As a direct corollary, the expansion yields the corresponding congruence relations and proves part(i) of the Chen–Liu–Zhu $SU(n)$ volume conjecture for $T(2,2p+1)$.

The classical theory of compact translation surfaces is organized through the stratification of moduli spaces according to the orders of the zeros of the associated Abelian differentials. In the setting of tame translation surfaces of infinite topological type, no analogous notion of strata has been systematically developed. In this article, we introduce a stratification theory for tame translation surfaces based on their singularity data. To each non-compact tame translation surface, we associate a sequence recording the cardinalities of finite-angle cone singularities of each possible cone angle together with the cardinality of the set of infinite-angle singularities. We call this sequence the singularity data of the surface. We study the realization problem for singularity data on the Loch Ness monster. Our first main result shows that there is no tame translation structure on the Loch Ness monster having only a finite number of finite cone angle singularities. We then prove that this is the only obstruction: every singularity data not excluded by the previous result is realized by a tame translation structure on the Loch Ness monster. As a consequence, we obtain a complete characterization of the strata of tame translation structures on this surface in terms of their singularity data. These results provide the first realization theorem for strata of non-compact tame translation surfaces and reveal a strong interaction between singularity data and the ends space.

Knot theory has provided a rich topological taxonomy for band structures, but its reach is fundamentally limited: knot invariants classify only 1D nodal lines at gap closure, and cannot encode the full dispersion or rich Fermi surface structure of realistic materials. Here we show that knotted graphs (knots that admit graph-like intersections in 3D space) - which have so far been elusive in condensed matter literature - provide a unified topological language for classifying the entire band dispersion, and even the eigenstate topology in some contexts. We propose a new framework beyond the existing Yamada polynomials that can topologically characterize the intricacies of realistic Fermi surfaces completely, crucially including how their multiple disconnected pieces are nested. This yields the Yamada set, a boundary-resolved extension which organizes the full topological evolution across energy into a Yamada sequence: a compact dispersion-level fingerprint directly tied to experimental signatures of Lifshitz transitions. Our framework is demonstrated with DFT-based band structures of real materials. Beyond dispersion-level classifications, this framework can be extended to non-Hermitian exceptional surfaces, where Berry-curvature flux further equips the knotted-graph skeleton with a directed Abelian edge flow that also captures the eigenstate topology.

We use reflection and rotation signatures of dihedral groups to develop a Smith normal form method for computing the conjugation quandle colorings across all dihedral groups. Let $L = L_1 \cup \cdots \cup L_\ell$ be a link with link components $L_1, \ldots, L_\ell$, and let $\operatorname{Conj}(D_n)$ be the conjugation quandle of the dihedral group $D_n$. We use the semidirect-product structure $D_n \cong \mathbb{Z}_n \rtimes \mathbb{Z}_2$ to separate the coloring into a component signature. This component signature tells us whether each link component is colored by rotations or reflections, and the exponents are assigned separately under modulus $n$. Once the component signature is determined, every crossing relation becomes an equation that determines an integer matrix $M_\tau$. The abelian group $A_\tau(L)$, which we call the signature module, is represented by an integer matrix $M_\tau$. We prove that the free rank and nonunit Smith normal form entries, or equivalently the isomorphism class of $A_\tau(L)$, are preserved under the Reidemeister moves. The complete family of signature modules also determines the entire family of counting invariants for all $D_n$. The all-reflection system is the same as the Fox-$n$ colorings, while the mixed rotation-reflection signatures contain information that gives a stronger invariant. We also show that each signature module is the $t_k = (-1)^{\tau_k}$ specialization of the multivariable Alexander module, and extend the coloring interpretation of the module to generalized dihedral groups $\operatorname{Dih}(B)$ for every abelian group $B$.

We prove that the knot Floer complex, along with some restrictions on the flip map, determines the dual knot to $\pm 1$ surgery on the Borromean knot. In particular, this implies that Heegaard Floer homology detects boundary Dehn twists.

Gabai proved that any Murasugi sum of $\pi_1$-essential Seifert surfaces is also $\pi_1$-essential, and Ozawa extended this result to unoriented spanning surfaces. We show, however, that the analogous statement about geometrically essential surfaces is untrue. (A spanning surface is geometrically essential if it cannot be compressed or boundary compressed to another spanning surface.) We also ask when plumbing an unknotted annulus (with any number of twists) onto a compressible spanning surface yields a compressible surface. We provide positive and negative examples, and we establish a simple sufficient condition. As an application, we obtain a new constructive proof of a result of Hatcher and Thurston about essential spanning surfaces for 2-bridge knots and links.

We prove that all Morita classes $\mu_k \in H_{4k}(Aut(F_{2k+2});\mathbb{Q})$ are nonzero using a novel description of the Hopf algebra structure on Steinberg homology. We prove analogous results with twisted coefficients. Our results disprove a case of a finiteness conjecture of Kontsevich.

We use the Labourie–Wentworth's formula and the thermodynamic formalism to show that, along the slice constructed by Bronstein–Davalo, the logarithmic top-eigenvalue length spectrum has a uniformly positive second variation near the Barbot representation. As a major application, we show that every closed surface group admits a non-Hitchin Borel Anosov representation into $\mathrm{SL}_{3k}\mathbb{R}$ for every $k\geqslant 1$. In particular, we obtain the first such examples in the even dimensions $6k$. We also study the local behavior of related objects of this slice around the Barbot representation, including the Lyapunov exponent of the flat bundle, the Hausdorff dimension of the limit set, and the Hilbert entropy of the representation.

We study the Gukov–Manolescu quantum invariant for oriented links. While this invariant is a single series in the knot case, we find that links admit multiple such series, each one consistent with the Melvin–Morton–Rozansky expansion of the colored Jones polynomials. We prove that convergent inverted state sums yield a family of multivariable Gukov–Manolescu series, indexed by monomials of the Alexander polynomial. We conjecture that these monomials correspond precisely to the fibered faces of the Thurston norm ball and provide extensive computational evidence for this correspondence. Further results concerning the leading term, the corresponding single-variable invariant, and the effect of partial Dehn surgery are also established.

We study how amphichirality of a knot constrains its braid index and, in the smallest nontrivial case, the combinatorics of its braid words. Using the Dynnikov–Prasolov resolution of the Jones conjecture, we show the braid index of an amphichiral knot is odd. This answers, in the negative, a question of Stoimenow on amphichiral knots of even braid index. We then classify prime amphichiral knots of braid index~$3$. Every amphichiral knot of braid index~$3$ is alternating and admits a minimal $3$-braid representative in a standard form encoded by a word~$c$. Building on the Birman–Menasco classification of closed $3$-braids and the Murasugi normal form, we describe a dihedral action on~$c$ under which the mirror and mirror-reverse operations are realized by a rotation and a reflection. For a standard form whose closure is a prime knot of braid index~$3$, this yields a complete criterion: $\widehat{\beta_c}$ is amphichiral if and only if $c$ is a palindrome or has odd period, together with a determination of the precise symmetry type in terms of these properties and the presence of a non-degenerate flype.

Let $X$ be a hyperbolic Riemann surface and let $\mu$ and $\nu$ be laminations on $X$ homotopic to measured foliations with finite Dirichlet integral. We prove that $\mu$ and $\nu$ are filling if and only if there exists a homeomorphism to another Riemann surface $f:X\to Y$ and an integrable holomorphic quadratic differential $q$ on $Y$, unique up to the natural equivalence, such that the push-forward laminations are homotopic to the horizontal and vertical foliations of $q$ respectively. This extends a classical theorem of Gardiner-Masur from closed surfaces to arbitrary surfaces. As well, the dual $\mathbb{R}$-tree interpretation yields the solution of an asymptotic Plateau problem for minimal surfaces in a product of two $\mathbb{R}$-trees. We construct examples such that $f:X\to Y$ is not homotopic to a quasiconformal map, and we present sufficient conditions that ensure it is. We deduce applications to main inequalities for locally quasiconformal maps, harmonic maps between surfaces, and big mapping class groups.

We give a general method for proving Kazhdan's property $(T)$ for quotients $G/K_{[c+1]}$, where $G$ is countable, $K$ is a normal subgroup and $K_{[j]}$ denotes its lower central series. The method combines an affine realization of $G/[K,K]$, a contraction argument, and permanence for nilpotent normal subgroups. We apply this method to several classes of groups by means of homological representations. For the Torelli lower-central series quotients $\operatorname{Mod}(\Sigma_g)/(\mathcal{T}_g)_{[c+1]}$ for $g\ge3$ we show that they have property $(T)$ for every $c\ge1$. We also prove property $(T)$ for $\operatorname{Aut}(F_3)/(\mathrm{IA}_3)_{[c+1]}$ for every $c\ge1$, relating these groups to the tame nilpotent images studied by Lubotzky and Pak. Several other classes of examples are discussed as well. We also prove that the Torelli lower-central series quotients for $c\ge2$ are not residually finite, and thus non-linear.

We prove that the critical value $2\pi^2$ is isolated for the Möbius cross energy of two-component links in the round three dimensional sphere $\mathbb{S}^{3}$. Specifically, there exists $\varepsilon_0>0$ such that any non-split, regular $H^2$ pair of curves with disjoint images, having vanishing first variation and energy at most $2\pi^2+\varepsilon_0$, must lie in the Möbius-reparametrization orbit of the standard Hopf link.

We prove the Chen–Yang volume conjecture for all sufficiently long integral Dehn fillings of any fixed marked fundamental shadow-link exterior. For each fixed filling, the exponential growth of its $SO(3)$ Turaev–Viro invariants recovers its hyperbolic volume along the full sequence of odd levels. The filling coefficients may have mixed signs and unrelated magnitudes. We also establish a complete asymptotic expansion of the signed $SO(3)$ Witten–Reshetikhin–Turaev invariant and identify the absolute leading coefficient explicitly in terms of adjoint Reidemeister torsion. Fixed even colors on the filling cores recover characters of the geometric holonomy. The key difficulty is cancellation in the signed surgery sum. Our main analytic tool transfers an exact reflection symmetry from a continuous model to the finite quantum sums. We control the error below the exponential scale of the surviving contribution. We also apply the method to a one-edge state sum restricted to central colors. The dominant contributions cancel, and for each sufficiently large fixed number of blocks we determine the smaller surviving exponential rate and its nonzero leading coefficient.

We extend the sphere-summing construction of \cite{AZ} from torus sums of spheres to symplectic sums along surfaces of arbitrary genus. The relative surfaces may have arbitrary positive intersection number with the summing surface. We give formulas for the genus and self-intersection of the resulting surface, together with graph and simultaneous versions of the construction. The original elliptic-surface case is recovered as a special case, and higher-genus examples are obtained from hyperelliptic Lefschetz fibrations and braided symplectic surfaces in $\Sigma_h\times S^2$.

The Katz-Long-Moody construction associates to a representation of the semidirect product of a free group and a braid group, defined by the Artin action, and a nonzero parameter a new representation of the same group. On the pure braid group it corresponds to Haraoka's multiplicative middle convolution for KZ-type equations. For unitary input and a convolution parameter on the unit circle other than one, the construction equips the quotient representation with a canonical non-degenerate invariant Hermitian form. We give a closed formula for its signature in terms of the eigenangles of the input local monodromies, the eigenangles of their ordered product, and the convolution parameter. The formula accounts for the kernel of the form before passage to the quotient and for signature changes at resonant parameters. It determines precisely when the induced form is definite, answering the definiteness problem posed in the companion paper. Definiteness implies unitarizability of the output representation; the converse holds when that representation is irreducible. The proof uses elementary linear algebra: a determinant identity, explicit block-pivot formulas, and an inertia formula for sums of Cayley transforms of unitary matrices. As applications, we compare the rank-one construction explicitly with Haraoka's invariant form for Pochhammer systems, recover the Gauss case of the Beukers-Heckman interlacing criterion, and determine the definite parameter intervals for the Hecke and Temperley-Lieb specializations.

We construct symplectic four-manifolds by taking mixed fiber sums along explicit cyclic multisections in ruled surfaces. For a connected unbranched degree-$p$ multisection in $\Sigma_g\times S^2$, we determine the first homology and fundamental group of the complement and prove that its boundary is incompressible. It follows that no direct gluing of two such complements can be simply connected; moreover, the first homology of every direct sum retains finite quotients determined by the covering degrees. We classify the mixed sums having Euler characteristic $4$ and signature $0$. Up to interchanging the two summands, exactly three possibilities occur, corresponding to the degree pairs $(2,3)$, $(2,4)$, and $(3,3)$. For each of these cases, suitable adapted product-framed symplectic gluings have the rational cohomology ring of $S^2\times S^2$. Varying the gluing by symplectic transvections produces infinitely many pairwise nondiffeomorphic examples, distinguished by the unbounded orders of their finite first homology groups. We also construct the twisted ruled analogue of the $(2,4)$ case. Explicit finite-holonomy multisections give connected square-zero symplectic surfaces in the classes $2S_3-F_3$ and $4S_2-2F_2$ in the nontrivial $S^2$-bundles over $\Sigma_3$ and $\Sigma_2$, respectively. More generally, for a square-zero degree-$p$ multisection in the nontrivial bundle the complement has first homology $\mathbb Z^{2g}\oplus\mathbb Z/(p/2)$. Suitable gluings in the twisted $(2,4)$ case have $b_1=0$, $b_2=2$, and signature zero, and every such sum is non-spin. Hence they have the rational cohomology ring of $\mathbb CP^2\#\overline{\mathbb CP}^{\,2}$. We compare these constructions with the author's 2006 construction of minimal symplectic four-manifolds having the integral cohomology $S^2\times S^2$, obtained via knot surgery and twisted fiber sums.

We classify commensurability relations between Deligne–Mostow–Thurston monodromy groups and the sixteen arithmetic Ghazouani–Pirio monodromy groups arising from moduli of flat cone metrics on the sphere and on the torus. In both settings, the monodromy group preserves a skew-Hermitian form coming from twisted homology. We use a degeneration method to compute their determinant classes in the genus one case. The defining CM fields and determinant classes of the skew-Hermitian forms determine the commensurability relations. In particular, each of the sixteen arithmetic Ghazouani–Pirio monodromy groups is commensurable with an arithmetic Deligne–Mostow–Thurston monodromy group.

We study representations of fundamental groups of closed orientable surfaces into \(\mathrm{SL}(2,\mathbb{H})\), whose projectivization \(\mathrm{PSL}(2,\mathbb{H})\) is the group of orientation-preserving isometries of real hyperbolic \(5\)-space, with the aim of developing a quaternionic analogue of Fenchel–Nielsen theory. We give a normal form for pairs of regular loxodromic elements with disjoint fixed point sets and show that, on an open dense generic locus, the corresponding moduli space is \(15\)-dimensional. We also associate fifteen word traces whose differentials are linearly independent on an open dense subset and hence give local real-analytic coordinates there. For a pair of pants with prescribed regular loxodromic boundary conjugacy classes, we show that the relative deformation space is locally \(6\)-dimensional. Combining this internal pants data with the three-dimensional boundary conjugacy data and the three-dimensional centralizer gluing freedom gives a local parameter decomposition for closed surface group representations. For a closed surface of genus \(g\), this yields \(30g-30\) real parameters.

Viro's Euler-integral polynomial $P_C(q)$ and the Lanzat–Polyak quantized-curvature polynomial $I_q(C)$ refine Arnold's invariants $J^-$ and $J^+$ for generic immersed one-component plane curves. We construct homological lifts of both. The bigraded region homology retains the singular homology of every connected Alexander-index region; its graded Euler characteristic is $P_C(q)$. The triply graded smoothing-circle homology is generated by the oriented circles of the orientation-preserving smoothing and decategorifies to the smoothing term in $I_q(C)$. Keeping the actual region summands and the boundary regions of every smoothing circle gives a homological refinement of the oriented smoothing configuration, or Seifert state. An infinite family proves strictness: both polynomial data and the ordinary homological lifts agree, while the component-graded region homology and the branch-decomposed circle homology distinguish every pair. Further constructions recover the full $I_q(C)$ by a vertex complex, realize the local change of its curvature integral by edge homology, and give a canonical two-state homology for unoriented curves. Viro described his Euler-integral formula as an analogue of face state-sum formulas for quantum knot polynomials. Through the categorifications developed here, we obtain one concrete homological face-state-sum model realizing that analogy.

For every $n\geq 3$ and $s\geq 2$, we construct a homogeneous polynomial of degree $n(n+1)/2$ whose Milnor fiber is exactly $2s$-connected and whose rational cohomology contains a strictly defined nontrivial $n$-fold Massey product on classes of degree $2s+1$, implying that the Milnor fiber is non-formal, while attaining the Kato–Matsumoto connectivity bound. Our construction is based on the simplicial multiwedges of the nerve complexes of simple polytopes introduced by Limonchenko, combined with Suciu's realization of weighted homogeneous Milnor fibers. We thereby answer two problems posed by Suciu.

We exhibit an annular knot with wrapping number four whose Kauffman bracket has annular degree at most two. This disproves the wrapping number conjecture.

We develop bar cohomology of link complements as an invariant of links in homology spheres. In this setting, bar cohomology is a Hopf algebra which is calculable using surfaces and their intersection curves in a link complement. In this first in a sequence of works, we introduce the invariant and show that it defines a canonical subspace of the tensor Hopf algebra, which already encodes information about Milnor's link invariants and provides geometrically significant information beyond them.

In this paper, we study the extensions of KLRW algebras to tensor products of Verma module representations of $\mathfrak{sl}_2$. Our motivation is to construct a theory of Khovanov homology for knot complements in $S^3$ (and which also categorifies the Gukov-Manolescu two-variable series for knot complements), which will be done in the second part of this work. We construct the categorification of R-matrices for Verma modules as functors given by derived tensor products with diagrammatic bimodules and explicitly compute their projective resolutions. We also prove these braiding functors induce an action of the braid group on the relevant categories. Then, we describe how to incorporate strands in finite-dimensional representations of $\mathfrak{sl}_2$, thereby establishing functors that serve as the Khovanov homology on a braid complement. In the case of the unknot, this gives knot homologies in $S^1\times D^2$, which we compare to Annular Khovanov Homology through several examples and show they are very closely related, conjecturing they are of the same dimension. We conclude with a proposal for the categorification of the cups and caps of Verma module colored strands, which we build upon in the next paper.

We study profinite rigidity of real hyperbolic lattices, with the main application in dimension three. The basic tool is a profinite version of the dual cube complex of a pair of filling curves on a surface, which we call a profinite cubical residue complex. We prove that an equivariant isomorphism between the completions of two such complexes is induced, after conjugation by a single element, by an isomorphism of the discrete groups and complexes, and that the same holds for free cocompact cellular actions of finitely generated residually finite groups on connected, locally finite, finite dimensional regular CW complexes. On a closed orientable surface of genus at least two, a theory of divisible fillings determines the completed cut tree of a nonseparating simple closed curve. Combined with the homological intersection criterion of Boggi and Zalesskii, this determines the completed dual square complex of a nonseparating filling pair, and hence realizes every isomorphism of profinite surface groups which preserves the two marked cyclic subgroups. In dimension three we combine these results with virtual fibering, Liu's profinite invariants of hyperbolic $3$-manifolds and the goodness of $3$-manifold groups. Massey products and virtual domination give cohomological integrality, and pseudo-Anosov dynamics, periodic orbit traces and cross sections of suspension flows give a correspondence of prime periodic orbits together with the fiber curves needed for the realization. Profinite rigidity of closed and of cusped lattices follows, the cusped case through cyclic orbifold fillings, and we obtain $\Out(\Gamma)\cong\Out(\wh\Gamma)$ for every lattice of $\PSL_2(\C)$.

We construct a $3$-generator $9$-relator group with unsolvable word problem. We use the group to construct two fixed-size Adian–Rabin families of group presentations, one with $4$ generators and $11$ relators, and another with $2$ generators and $10$ relators. As a consequence, $\#_7(S^2\times S^2)$ is topologically unrecognizable and $\#_9(S^2\times S^2)$ is smoothly unrecognizable. These algebraic and topological results improve the previous best known bounds by Borisov, Tancer, and Gordon. The construction of the group builds upon an example of Borisov and uses additional HNN extensions and Tietze eliminations to reduce the size of the presentation. We also provide a machine-checked Lean~4 formalization of the algebraic results.

We investigate the capability of medial quandles to detect causality in (2+1) dimensional globally hyperbolic spacetime by determining if their coloring link invariants can distinguish between the connected sum of two Hopf links and an infinite series of relevant three-component links constructed by Allen and Swenberg in 2020, who suggested that any link invariant must be able to distinguish those links for them to detect causality in the given setting. We show that these quandles fail to do so as long as $a\sim b\Leftrightarrow a*b=a$ defines an equivalence relation. The Alexander quandles are an example to which this result applies. We also give a sufficient condition for a medial quandle to distinguish between the connected sum of two Hopf links and all links in Allen-Swenberg series, and give an example of such a quandle. Inspired by this result, we also derive a generalized theorem about the coloring of medial quandles on a specific type of tangle, which helps to determine whether these quandles can distinguish between a wider range of knots and links or not.

We prove that every simplicial triangulation of real projective $d$-space has $\exp(\Omega(\sqrt d))$ vertices. Together with known constructions, this determines the minimum vertex number as $\mu_d=\exp(d^{1/2+o(1)})$. The result follows from a topological generalization of the Figiel–Lindenstrauss–Milman inequality, answering a recent question of Frick, Hosseini, and Vasileuski: a finite strongly regular CW complex with a free cellular involution, $v$ vertices, and $f$ maximal cells has $\mathbb{Z}/2$-index at most $O(\log v\log f)$. We bound the dimensions of Morse cells by a trace estimate for a constrained Hessian, obtaining a Morse-theoretic proof of the classical inequality for centrally symmetric polytopes. As further applications of this inequality, we give an $\exp(\Omega(\sqrt t))$ lower bound for the order of a triangle-free topologically $t$-chromatic graph and bound the index of sign complexes by $O(d\log^2 N)$ for total matrices and $O(d\log^3 N)$ for partial matrices, where $N\geq2$ is the number of columns and $d\geq1$ is the VC dimension.

In this paper, we count the number of (almost) extremally stretched closed geodesics for a pair of (non-conjugate) Fuchsian representations of a closed surface group, and show that it has subexponential growth. We then deduce that, as a discrete subgroup of $\mathsf{PO}(2, 1) \times \mathsf{PO}(2, 1)$, the growth indicator of the product representation vanishes on the boundary of the Benoist limit cone. We also prove that for a general Zariski dense Borel Anosov subgroup of $\mathsf{PO}(2, 1) \times \mathsf{PO}(2, 1)$, its growth indicator vanishes on at least one boundary component of the Benoist limit cone, but not necessarily on both. One may view our first result as a sharpening of Thurston's result that there is a unique geodesic lamination $\lambda$ such that every measured lamination maximizing the ratio of lengths with respect to the two representations has support contained in $\lambda$. We hope this will be a starting point for a more general study of extremal entropy.

We study the Hopf decomposition of subgroup actions of the Teichmüller modular group on the Thurston boundary with respect to the Thurston measure class. We identify the conservative part with the big horospherical limit set modulo null sets. For a basepoint with trivial stabilizer in the subgroup, the ideal boundary of the associated Dirichlet polyhedron is wandering, and its subgroup translates cover the dissipative part modulo null sets. The dissipative part also agrees with the set of Dirichlet points modulo null sets. The description using Dirichlet polyhedra relies on a separation theorem for extremal length: every level set of an extremal length ratio at distinct points of Teichmüller space has measure zero. Kaimanovich's Radon–Nikodym criterion characterizes the two parts by the divergence and convergence, respectively, of a series of extremal length ratios. For the Torelli group of a closed surface of genus at least two, we use radial limits of the period map to prove that its conical limit set has measure zero. Together with the conservativity established by Choi, Gekhtman, Yang, and Zheng, our geometric characterization implies that its big horospherical limit set has full measure and that the ideal boundary of every Dirichlet polyhedron has measure zero.