IA Scholar Query: Zero-Parity Stabbing Information
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgTue, 29 Nov 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Weights of mod p automorphic forms and partial Hasse invariants
https://scholar.archive.org/work/vjj6htmeabfdxb7o4bjtjkh75i
For a connected, reductive group G over a finite field endowed with a cocharacter μ, we define the zip cone of (G,μ) as the cone of all possible weights of mod p automorphic forms on the stack of G-zips. This cone is conjectured to coincide with the cone of weights of characteristic p automorphic forms for Hodge-type Shimura varieties of good reduction. We prove in full generality that the cone of weights of characteristic 0 automorphic forms is contained in the zip cone, which gives further evidence to this conjecture. Furthermore, we determine exactly when the zip cone is generated by the weights of partial Hasse invariants, which is a group-theoretical generalization of a result of Diamond–Kassaei and Goldring–Koskivirta.Naoki Imai, Jean-Stefan Koskivirtawork_vjj6htmeabfdxb7o4bjtjkh75iTue, 29 Nov 2022 00:00:00 GMTSymmetric Formulas for Products of Permutations
https://scholar.archive.org/work/vfh5lxdmqbdure7o6pfw5kgx4e
We study the formula complexity of the word problem 𝖶𝗈𝗋𝖽_S_n,k : {0,1}^kn^2→{0,1}: given n-by-n permutation matrices M_1,...,M_k, compute the (1,1)-entry of the matrix product M_1⋯ M_k. An important feature of this function is that it is invariant under action of S_n^k-1 given by (π_1,...,π_k-1)(M_1,...,M_k) = (M_1π_1^-1,π_1M_2π_2^-1,...,π_k-2M_k-1π_k-1^-1,π_k-1M_k). This symmetry is also exhibited in the smallest known unbounded fan-in {𝖠𝖭𝖣,𝖮𝖱,𝖭𝖮𝖳}-formulas for 𝖶𝗈𝗋𝖽_S_n,k, which have size n^O(log k). In this paper we prove a matching n^Ω(log k) lower bound for S_n^k-1-invariant formulas computing 𝖶𝗈𝗋𝖽_S_n,k. This result is motivated by the fact that a similar lower bound for unrestricted (non-invariant) formulas would separate complexity classes 𝖭𝖢^1 and 𝖫𝗈𝗀𝗌𝗉𝖺𝖼𝖾. Our more general main theorem gives a nearly tight n^d(k^1/d-1) lower bound on the G^k-1-invariant depth-d {𝖬𝖠𝖩,𝖠𝖭𝖣,𝖮𝖱,𝖭𝖮𝖳}-formula size of 𝖶𝗈𝗋𝖽_G,k for any finite simple group G whose minimum permutation representation has degree n. We also give nearly tight lower bounds on the G^k-1-invariant depth-d {𝖠𝖭𝖣,𝖮𝖱,𝖭𝖮𝖳}-formula size in the case where G is an abelian group.William He, Benjamin Rossmanwork_vfh5lxdmqbdure7o6pfw5kgx4eMon, 28 Nov 2022 00:00:00 GMTUnfolding, higher spins, metaplectic groups and resolution of classical singularities
https://scholar.archive.org/work/bakpa5ajw5d75edzfistuzdgsa
We review and extend some recent results concerning the analysis of spacetime singularities in four-dimensional higher spin gravity, summarizing how the coupling of the gravitational field to massless higher spins may provide resolution mechanisms. We elucidate such mechanisms at the level of curvature singularities and degenerate metrics in exact as well as linearized solutions to Vasiliev's equations. As a preamble, we review the underlying higher-spin algebra and its metaplectic group extensions, after which we detail various gauge functions encoding the 𝐴𝑑𝑆 4 vacuum and the non-rotating Bañados-Gomberoff-Martinez (BGM) metric, the four-dimensional lift of the spinless BTZ black hole, in different coordinate patches related by transition functions. We then revisit how, within the unfolded formalism, it is natural extend the BGM black hole through its causal singularity. Finally, we compare the metric-like and unfolded descriptions of scalar fluctuations over the (extended) BGM background, showing how the latter description maps singularities to well-defined metaplectic group elements providing regular values for the Weyl zero-form master field, which thus admits continuation over the full extended BGM spacetime.Carlo Iazeolla, Per Sundellwork_bakpa5ajw5d75edzfistuzdgsaWed, 23 Nov 2022 00:00:00 GMTGalois closure of a 5-fold covering and decomposition of its Jacobian
https://scholar.archive.org/work/z5k6mcacgrbw5piqzzd7jrhfwy
For an arbitrary 5-fold ramified covering between compact Riemann surfaces, every possible Galois closure is determined in terms of the ramification data of the map; namely, the ramification divisor of the covering map. Since the group that acts on the Galois closure also acts on the Jacobian variety of the covering surface, we describe its group algebra decomposition in terms of the Jacobian and Prym varieties of the intermediate coverings of the Galois closure. The dimension and induced polarization of each abelian variety in the decomposition is computed in terms of the ramification data of the covering map.Benjamín M. Moragawork_z5k6mcacgrbw5piqzzd7jrhfwyMon, 21 Nov 2022 00:00:00 GMTLectures on modular forms and strings
https://scholar.archive.org/work/rra4lqiauzezpnt4kq5ag3axnu
The goal of these lectures is to present an informal but precise introduction to a body of concepts and methods of interest in number theory and string theory revolving around modular forms and their generalizations. Modular invariance lies at the heart of conformal field theory, string perturbation theory, Montonen-Olive duality, Seiberg-Witten theory, and S-duality in Type IIB superstring theory. Automorphic forms with respect to higher arithmetic groups as well as mock modular forms enter in toroidal string compactifications and the counting of black hole microstates. After introducing the basic mathematical concepts including elliptic functions, modular forms, Maass forms, modular forms for congruence subgroups, vector-valued modular forms, and modular graph forms, we describe a small subset of the countless applications to problems in Mathematics and Physics, including those mentioned above.Eric D'Hoker, Justin Kaidiwork_rra4lqiauzezpnt4kq5ag3axnuFri, 18 Nov 2022 00:00:00 GMTOn minimal tilting complexes in highest weight categories
https://scholar.archive.org/work/6wql43voy5ck5bzuam6jrtw2ee
We explain the construction of minimal tilting complexes for objects of highest weight categories and we study in detail the minimal tilting complexes for standard objects and simple objects. For certain categories of representations of complex simple Lie algebras, affine Kac-Moody algebras and quantum groups at roots of unity, we relate the multiplicities of indecomposable tilting objects appearing in the terms of these complexes to the coefficients of Kazhdan-Lusztig polynomials.Jonathan Gruberwork_6wql43voy5ck5bzuam6jrtw2eeFri, 18 Nov 2022 00:00:00 GMTTweeting 'truths': rumour and grammars of power in Kenya
https://scholar.archive.org/work/7em2csjco5gcnblhxrn6musg7u
This study examines rumour as a mediator of public discourses in Kenya. It focuses on rumours that followed the killing of Chris Msando – a senior election official with the Independent Electoral and Boundaries Commission – and his friend Carolyne Ngumbu a week before the 2017 elections. Although earlier research on rumour exists, it is limited to oral societies and overlooks the versatility of structure and functions of rumours. Therefore, I study the interface between rumours, Twitter and the politics surrounding the two deaths. The research is informed by four objectives: to trace the history of rumour as an area of study in Africa; evaluate the role of Twitter in the creation, circulation, and use of rumour in contemporary Kenya; discuss the uses of rumour for government and individuals; and analyse how the interface between rumours and Twitter impact on the everyday life in Kenya. I use close textual reading of rumours and informal conversations to corroborate data scraped from Twitter. I then apply four theories: Fairclough's Critical Discourse Analysis (1995) to unpack the meanings in rumours and informal conversations; Paul Ricoeur's (1973) notion of hermeneutics of suspicion as popularised by Felski (2011) to analyse the rumours; Deleuze and Guattari's (1987) metaphorical rhizome to understand the amorphousness of rumours; and Jodi Dean's (2009) concept of communicative capitalism to determine the extent to which communicative technologies of popular and social media's appropriation in rumourous conversations evoke political awareness among different interlocutors. This thesis argues that rumours have changed and been changed by Twitter's communicative cultures, owing to their structural complexity and the growing alertness among the general publics about the necessity of self-expression in a country where a majority of citizens have accepted democracy as the most desirable basis of political organization. Thus, contemporary rumours emerge from the process of co-creation in an amorphous public struggling [...]Denis Galava, University Of Edinburgh, Thomas Molony, Paul Nugentwork_7em2csjco5gcnblhxrn6musg7uThu, 17 Nov 2022 00:00:00 GMTBorder complexity via elementary symmetric polynomials
https://scholar.archive.org/work/sz7qht3eszewzayzsbpyuoc3mu
In (ToCT'20) Kumar surprisingly proved that every polynomial can be approximated as a sum of a constant and a product of linear polynomials. In this work, we prove the converse of Kumar's result which ramifies in a surprising new formulation of Waring rank and border Waring rank. From this conclusion, we branch out into two different directions, and implement the geometric complexity theory (GCT) approach in two different settings. In the first direction, we study the orbit closure of the product-plus-power polynomial, determine its stabilizer, and determine the properties of its boundary points. We also connect its fundamental invariant to the Alon-Tarsi conjecture on Latin squares, and prove several exponential separations between related polynomials contained in the affine closure of product-plus-product polynomials. We fully implement the GCT approach and obtain several equations for the product-plus-power polynomial from its symmetries via representation theoretic multiplicity obstructions. In the second direction, we demonstrate that the non-commutative variant of Kumar's result is intimately connected to the constructions of Ben-Or and Cleve (SICOMP'92), and Bringmann, Ikenmeyer, Zuiddam (JACM'18), which describe algebraic formulas in terms of iterated matrix multiplication. From this we obtain that a variant of the elementary symmetric polynomial is complete for V3F, a large subclass of VF, under homogeneous border projections. In the regime of quasipolynomial complexity, our polynomial has the same power as the determinant or as arbitrary circuits, i.e., VQP. This is the first completeness result under homogeneous projections for a subclass of VBP. Such results are required to set up the GCT approach in a way that avoids the no-go theorems of B\"urgisser, Ikenmeyer, Panova (JAMS'19). ...Pranjal Dutta and Fulvio Gesmundo and Christian Ikenmeyer and Gorav Jindal and Vladimir Lysikovwork_sz7qht3eszewzayzsbpyuoc3muMon, 14 Nov 2022 00:00:00 GMTQuantum error mitigation for rotation symmetric bosonic codes with symmetry expansion
https://scholar.archive.org/work/k2w72lrwdffehlcw2kfctd46ba
The rotation symmetric bosonic code (RSBC) is a unified framework of practical bosonic codes that have rotation symmetries, such as cat codes and binomial codes. While cat codes achieve the break-even point in which the coherence time of the encoded qubits exceeds that of unencoded qubits, with binomial codes nearly approaching that point, the state preparation fidelity needs to be still improved for practical quantum computing. Concerning this problem, we investigate the framework of symmetry expansion, a class of quantum error mitigation that virtually projects the state onto the noise-free symmetric subspace by exploiting the system's intrinsic symmetries and post-processing of measurement outcomes. Although symmetry expansion has been limited to error mitigation of quantum states immediately before measurement, we successfully generalize symmetry expansion for state preparation. To implement our method, we use an ancilla qubit and only two controlled-rotation gates via dispersive interactions between the bosonic code states and the ancilla qubit. Interestingly, this method also allows us to virtually prepare the RSBC states only from easy-to-prepare states, e.g., coherent states. We also discuss that the conventional symmetry expansion protocol can be applied to improve the computation fidelity when the symmetries of rotation bosonic codes are unavailable due to low measurement fidelity. By giving comprehensive analytical and numerical arguments regarding the trace distance between the error-mitigated state and the ideal state and the sampling cost of quantum error mitigation, we show that symmetry expansion dramatically suppresses the effect of photon loss. Our novel error mitigation method will significantly enhance computation accuracy in the near-term bosonic quantum computing paradigm.Suguru Endo, Yasunari Suzuki, Kento Tsubouchi, Rui Asaoka, Kaoru Yamamoto, Yuichiro Matsuzaki, Yuuki Tokunagawork_k2w72lrwdffehlcw2kfctd46baFri, 11 Nov 2022 00:00:00 GMTTensor diagrams and cluster combinatorics at punctures
https://scholar.archive.org/work/mos5oumlwfgddbfxozsf25fm6q
Fock and Goncharov introduced a family of cluster algebras associated with the moduli of SL(k)-local systems on a marked surface with extra decorations at marked points. We study this family from an algebraic and combinatorial perspective, emphasizing the structures which arise when the surface has punctures. When k is 2, these structures are the tagged arcs and tagged triangulations of Fomin, Shapiro, and Thurston. For higher k, the tagging of arcs is replaced by a Weyl group action at punctures discovered by Goncharov and Shen. We pursue a higher analogue of a tagged triangulation in the language of tensor diagrams, extending work of Fomin and the second author, and we formulate skein-algebraic tools for calculating in these cluster algebras. We analyze the finite mutation type examples in detail.Chris Fraser, Pavlo Pylyavskyywork_mos5oumlwfgddbfxozsf25fm6qThu, 10 Nov 2022 00:00:00 GMTIntegral geometry on the octonionic plane
https://scholar.archive.org/work/cejhvrtx7vadhg5bpotklieqi4
We describe explicitly the algebra of Spin(9)-invariant, translation-invariant, continuous valuations on the octonionic plane. Namely, we present a basis in terms of invariant differential forms and determine the Bernig-Fu convolution on this space. The main technical ingredient we introduce is an extension of the invariant theory of the Lie group Spin(7) to the isotropy representation of the action of Spin(9) on the 15-dimensional sphere, reflecting the underlying octonionic structure. As an application, we compute the principal kinematic formula on the octonionic plane and express in our basis certain Spin(9)-invariant valuations introduced previously by Alesker.Jan Kotrbatý, Thomas Wannererwork_cejhvrtx7vadhg5bpotklieqi4Thu, 10 Nov 2022 00:00:00 GMTSigned permutohedra, delta-matroids, and beyond
https://scholar.archive.org/work/2yzqr44rpvgeheytlc7pyjivwi
We establish a connection between the algebraic geometry of the type B permutohedral toric variety and the combinatorics of delta-matroids. Using this connection, we compute the volume and lattice point counts of type B generalized permutohedra. Applying tropical Hodge theory to a new framework of "tautological classes of delta-matroids," modeled after certain vector bundles associated to realizable delta-matroids, we establish the log-concavity of a Tutte-like invariant for a broad family of delta-matroids that includes all realizable delta-matroids. Our results include new log-concavity statements for all (ordinary) matroids as special cases.Christopher Eur, Alex Fink, Matt Larson, Hunter Spinkwork_2yzqr44rpvgeheytlc7pyjivwiSun, 06 Nov 2022 00:00:00 GMTStability Conditions for 3-fold Flops
https://scholar.archive.org/work/u5cdxmj2d5g4pgueqggwr6lcly
Let f X→Spec R be a 3-fold flopping contraction, where X has at worst Gorenstein terminal singularities and R is complete local. We describe the space of Bridgeland stability conditions on the null subcategory 𝒞 of the bounded derived category of X, which consists of those complexes that derive pushforward to zero, and also on the affine subcategory 𝒟, which consists of complexes supported on the exceptional locus. We show that a connected component of stability conditions on 𝒞 is the universal cover of the complexified complement of the real hyperplane arrangement associated to X via the Homological MMP, and more generally that a connected component of normalised stability conditions on 𝒟 is a regular covering space of the infinite hyperplane arrangement constructed in Iyama-Wemyss [IW9]. Neither arrangement is Coxeter in general. As a consequence, we give the first description of the Stringy Kähler Moduli Space (SKMS) for all smooth irreducible 3-fold flops. The answer is surprising: we prove that the SKMS is always a sphere, minus either 3, 4, 6, 8, 12 or 14 points, depending on the length of the curve.Yuki Hirano, Michael Wemysswork_u5cdxmj2d5g4pgueqggwr6lclyWed, 02 Nov 2022 00:00:00 GMTSelf-Defense for Theists
https://scholar.archive.org/work/7pgz462fhrd4dh7znwecuhhly4
According to Theistic Defensive Incompatibilism, common theistic commitments limit the scope or explanation of permissible self-defense. In this essay, I offer six original arguments for Theistic Defensive Incompatibilism. The first four arguments concern narrow proportionality: the requirement that the defensive harm inflicted on unjust threateners not exceed the harm they threaten. Hellism, Annihilationism, and Danteanism each imply that narrow proportionality is rarely satisfied, whereas Universalism implies that killing never harms. The final two arguments concern wide proportionality, or the requirement that defensive harm not excessively harm non-liable third parties. Omnisubjectivity and Divine Love imply that wide proportionality is rarely satisfied.Blake Herethwork_7pgz462fhrd4dh7znwecuhhly4Fri, 21 Oct 2022 00:00:00 GMTLearning to Reason with Neural Networks: Generalization, Unseen Data and Boolean Measures
https://scholar.archive.org/work/ntvwk2x7vjc5dfq2dva4w3mkvm
This paper considers the Pointer Value Retrieval (PVR) benchmark introduced in [ZRKB21], where a 'reasoning' function acts on a string of digits to produce the label. More generally, the paper considers the learning of logical functions with gradient descent (GD) on neural networks. It is first shown that in order to learn logical functions with gradient descent on symmetric neural networks, the generalization error can be lower-bounded in terms of the noise-stability of the target function, supporting a conjecture made in [ZRKB21]. It is then shown that in the distribution shift setting, when the data withholding corresponds to freezing a single feature (referred to as canonical holdout), the generalization error of gradient descent admits a tight characterization in terms of the Boolean influence for several relevant architectures. This is shown on linear models and supported experimentally on other models such as MLPs and Transformers. In particular, this puts forward the hypothesis that for such architectures and for learning logical functions such as PVR functions, GD tends to have an implicit bias towards low-degree representations, which in turn gives the Boolean influence for the generalization error under quadratic loss.Emmanuel Abbe, Samy Bengio, Elisabetta Cornacchia, Jon Kleinberg, Aryo Lotfi, Maithra Raghu, Chiyuan Zhangwork_ntvwk2x7vjc5dfq2dva4w3mkvmThu, 20 Oct 2022 00:00:00 GMTBPS Dendroscopy on Local ℙ^2
https://scholar.archive.org/work/t6tg2abdljblvfhway6q7bxfuq
The spectrum of BPS states in type IIA string theory compactified on a Calabi-Yau threefold famously jumps across codimension-one walls in complexified Kähler moduli space, leading to an intricate chamber structure. The Split Attractor Flow Conjecture posits that the BPS index Ω_z(γ) for given charge γ and moduli z can be reconstructed from the attractor indices Ω_*(γ_i) counting BPS states of charge γ_i in their respective attractor chamber, by summing over a finite set of decorated rooted flow trees known as attractor flow trees. If correct, this provides a classification (or dendroscopy) of the BPS spectrum into different topologies of nested BPS bound states, each having a simple chamber structure. Here we investigate this conjecture for the simplest, albeit non-compact, Calabi-Yau threefold, namely the canonical bundle over P^2. Since the Kähler moduli space has complex dimension one and the attractor flow preserves the argument of the central charge, attractor flow trees coincide with scattering sequences of rays in a two-dimensional slice of the scattering diagram in the space of stability conditions on the derived category of compactly supported coherent sheaves on K_P^2. We combine previous results on the scattering diagram of K_P^2 in the large volume slice with new results near the orbifold point ℂ^3/ℤ_3, and argue that the Split Attractor Flow Conjecture holds true on the physical slice of Π-stability conditions. In particular, while there is an infinite set of initial rays related by the group Γ_1(3) of auto-equivalences, only a finite number of possible decompositions γ=∑_iγ_i contribute to the index Ω_z(γ) for any γ and z, with constituents γ_i related by spectral flow to the fractional branes at the orbifold point.Pierrick Bousseau, Pierre Descombes, Bruno Le Floch, Boris Piolinework_t6tg2abdljblvfhway6q7bxfuqWed, 19 Oct 2022 00:00:00 GMTThe Hodge bundle, the universal 0-section, and the log Chow ring of the moduli space of curves
https://scholar.archive.org/work/mbckeelrb5d6poq2mrq4gjugde
We bound from below the complexity of the top Chern class of the Hodge bundle in the Chow ring of the moduli space of curves: no formulas in terms of classes of degrees 1 and 2 can exist. As a consequence of the Torelli map, the 0-section over the second Voronoi compactification of the moduli of principally polarized abelian varieties also can not be expressed in terms of classes of degree 1 and 2. Along the way, we establish new cases of Pixton's conjecture for tautological relations. In the log Chow ring of the moduli space of curves, however, we prove that the top Chern class of the Hodge bundle lies in the subalgebra generated by logarithmic boundary divisors. The proof is effective and uses Pixton's double ramification cycle formula together with a foundational study of the tautological ring defined by a normal crossings divisor. The results open the door to the search for simpler formulas for the top Chern class of the Hodge bundle on the moduli of curves after log blow-ups.Samouil Molcho, Rahul Pandharipande, Johannes Schmittwork_mbckeelrb5d6poq2mrq4gjugdeMon, 17 Oct 2022 00:00:00 GMTThe monodromy of families of subvarieties on abelian varieties
https://scholar.archive.org/work/hbou346jgzfktpff6zsk2sv7y4
Motivated by recent work of Lawrence-Venkatesh and Lawrence-Sawin, we show that non-isotrivial families of subvarieties in abelian varieties have big monodromy when twisted by generic rank one local systems. While Lawrence-Sawin discuss the case of subvarieties of codimension one, our results hold for subvarieties of codimension at least half the dimension of the ambient abelian variety. For the proof, we use a combination of geometric arguments and representation theory to show that the Tannaka groups of intersection complexes on such subvarieties are big.Ariyan Javanpeykar, Thomas Krämer, Christian Lehn, Marco Maculanwork_hbou346jgzfktpff6zsk2sv7y4Tue, 11 Oct 2022 00:00:00 GMTDepth lower bounds in Stabbing Planes for combinatorial principles
https://scholar.archive.org/work/dm7upsei3fcwdbauiodonc5a6q
Stabbing Planes (also known as Branch and Cut) is a proof system introduced very recently which, informally speaking, extends the DPLL method by branching on integer linear inequalities instead of single variables. The techniques known so far to prove size and depth lower bounds for Stabbing Planes are generalizations of those used for the Cutting Planes proof system established via communication complexity arguments. As such they work for the lifted version of combinatorial statements. Rank lower bounds for Cutting Planes are also obtained by geometric arguments called protection lemmas. In this work we introduce two new geometric approaches to prove size/depth lower bounds in Stabbing Planes working for any formula: (1) the antichain method, relying on Sperner's Theorem and (2) the covering method which uses results on essential coverings of the boolean cube by linear polynomials, which in turn relies on Alon's combinatorial Nullenstellensatz. We demonstrate their use on classes of combinatorial principles such as the Pigeonhole principle, the Tseitin contradictions and the Linear Ordering Principle. By the first method we prove almost linear size lower bounds and optimal logarithmic depth lower bounds for the Pigeonhole principle and analogous lower bounds for the Tseitin contradictions over the complete graph and for the Linear Ordering Principle. By the covering method we obtain a superlinear size lower bound and a logarithmic depth lower bound for Stabbing Planes proof of Tseitin contradictions over a grid graph.Stefan Dantchev, Nicola Galesi, Abdul Ghani, Barnaby Martinwork_dm7upsei3fcwdbauiodonc5a6qSun, 09 Oct 2022 00:00:00 GMTGeodesic planes in a geometrically finite end and the halo of a measured lamination
https://scholar.archive.org/work/pfufyngipjdctixfgu5nsrjt7y
Recent works [MMO1, arXiv:1802.03853, arXiv:1802.04423, arXiv:2101.08956] have shed light on the topological behavior of geodesic planes in the convex core of a geometrically finite hyperbolic 3-manifolds M of infinite volume. In this paper, we focus on the remaining case of geodesic planes outside the convex core of M, giving a complete classification of their closures in M. In particular, we show that the behavior is different depending on whether exotic roofs exist or not. Here an exotic roof is a geodesic plane contained in an end E of M, which limits on the convex core boundary ∂ E, but cannot be separated from the core by a support plane of ∂ E. A necessary condition for the existence of exotic roofs is the existence of exotic rays for the bending lamination. Here an exotic ray is a geodesic ray that has finite intersection number with a measured lamination ℒ but is not asymptotic to any leaf nor eventually disjoint from ℒ. We establish that exotic rays exist if and only if ℒ is not a multicurve. The proof is constructive, and the ideas involved are important in the construction of exotic roofs. We also show that the existence of geodesic rays satisfying a stronger condition than being exotic, phrased in terms of only the hyperbolic surface ∂ E and the bending lamination, is sufficient for the existence of exotic roofs. As a result, we show that geometrically finite ends with exotic roofs exist in every genus. Moreover, in genus 1, when the end is homotopic to a punctured torus, a generic one (in the sense of Baire category) contains uncountably many exotic roofs.Tina Torkaman, Yongquan Zhangwork_pfufyngipjdctixfgu5nsrjt7ySat, 08 Oct 2022 00:00:00 GMT