Publications

1. Non-planar ends are continuously unforgettable.

   Preprint (2024).

   Joint with Javier Aramayona, Rodrigo De Pool, Rachel Skipper, Jing Tao, and Xiaolei Wu

   abstract bibtex arXiv

   @Misc{arXiv:2409.05502,
       Author = {Aramayona, Javier and De Pool, Rodrigo and Skipper, Rachel and Tao, Jing and Vlamis, Nicholas G. and Wu, Xiaolei},
       Title = {Non-planar ends are continuously unforgettable},
       Year = {2024},
       HowPublished = {Preprint, {arXiv}:2409.05502 [math.{GT}] (2024)},
       URL = {https://arxiv.org/abs/2409.05502},
       arXiv = {arXiv:2409.05502}
   }
   

   We show that continuous epimorphisms between a class of subgroups of mapping class groups of orientable infinite-genus 2-manifolds with no planar ends are always induced by homeomorphisms. This class of subgroups includes the pure mapping class group, the closure of the compactly supported mapping classes, and the full mapping class group in the case that the underlying manifold has a finite number of ends or is perfectly self-similar. As a corollary, these groups are Hopfian topological groups.
2. Covers of surfaces.

   Preprint (2024).

   Joint with Ian Biringer, Yassin Chandran, Tommaso Cremaschi, Jing Tao, Mujie Wang, and Brandis Whitfield

   abstract bibtex arXiv

   @Misc{arXiv:2409.03967,
       Author = {Biringer, Ian and Chandran, Yassin and Cremaschi, Tommaso and Tao, Jing and Vlamis, Nicholas G. and Wang, Mujie and Whitfield, Brandis},
       Title = {Covers of surfaces},
       Year = {2024},
       HowPublished = {Preprint, {arXiv}:2409.03967 [math.{GT}] (2024)},
       URL = {https://arxiv.org/abs/2409.03967},
       arXiv = {arXiv:2409.03967}
   }
   

   We study the homeomorphism types of certain covers of (always orientable) surfaces, usually of infinite-type. We show that every surface with non-abelian fundamental group is covered by every noncompact surface, we identify the universal abelian covers and the ℤ/nℤ-homology covers of surfaces, and we show that non-locally finite characteristic covers of surfaces have four possible homeomorphism types.
3. Homeomorphism groups of telescoping 2-manifolds are strongly distorted.

   Groups Geom. Dyn., to appear. Preprint (2024).

   abstract bibtex arXiv

   @Misc{arXiv:2403.03887,
       Author = {Vlamis, Nicholas G.},
       Title = {Homeomorphism groups of telescoping 2-manifolds are strongly distorted},
       Year = {2024},
       HowPublished = {Preprint, {arXiv}:2403.03887 [math.{GT}] (2024)},
       URL = {https://arxiv.org/abs/2403.03887},
       arXiv = {arXiv:2403.03887}
   }
   

   Building on the work of Mann and Rafi, we introduce an expanded definition of a telescoping 2-manifold and proceed to study the homeomorphism group of a telescoping 2-manifold. Our main result shows that it is strongly distorted. We then give a simple description of its commutator subgroup, which is index one, two, or four depending on the topology of the manifold. Moreover, we show its commutator subgroup is uniformly perfect with commutator width at most two, and we give a family of uniform normal generators for its commutator subgroup. As a consequence of the latter result, we show that for an (orientable) telescoping 2-manifold, every (orientation-preserving) homeomorphism can be expressed as a product of at most 17 conjugate involutions. Finally, we provide analogous statements for mapping class groups.
4. Orientation-preserving homeomorphisms of Euclidean space are commutators.

   Preprint (2024).

   Joint with Megha Bhat

   abstract bibtex arXiv

   @Misc{arXiv:2311.00229,
       Author = {Bhat, Megha and Vlamis, Nicholas G.},
       Title = {Orientation-preserving homeomorphisms of {Euclidean} space are commutators},
       Year = {2023},
       HowPublished = {Preprint, {arXiv}:2311.00229 [math.{GT}] (2023)},
       URL = {https://arxiv.org/abs/2311.00229},
       arXiv = {arXiv:2311.00229}
   }
   

   We prove that every orientation-preserving homeomorphism of Euclidean space can be expressed as a commutator. We establish an analogous result for annuli.
5. Homeomorphism groups of 2-manifolds with the virtual Rokhlin property.

   J. Topol., to appear. Preprint (2024).

   Joint with Justin Lanier

   abstract bibtex arXiv journal

   @Misc{arXiv:2308.13138,
       Author = {Lanier, Justin and Vlamis, Nicholas G.},
       Title = {Homeomorphism groups of 2-manifolds with the virtual {Rokhlin} property},
       Year = {2023},
       HowPublished = {Preprint, {arXiv}:2308.13138 [math.{GT}] (2023)},
       Keywords = {57K20,54H11,37B05,20E45},
       URL = {https://arxiv.org/abs/2308.13138},
       arXiv = {arXiv:2308.13138}
   }
   

   We introduce and motivate the definition of the virtual Rokhlin property for topological groups. We then classify the 2-manifolds whose homeomorphism groups have the virtual Rokhlin property. We also establish the analogous result for mapping class groups of 2-manifolds.
6. Homeomorphism groups of self-similar 2-manifolds.

   In the tradition of Thurston III. Geometry and Dynamics.

   Keni'chi Ohshika and Athanase Papadopoulos, eds., Springer, 2024, Chapter 5, 105--167.

   abstract bibtex arXiv journal

   @InCollection{zbMATH07918552,
       Author = {Vlamis, Nicholas G.},
       Title = {Homeomorphism groups of self-similar 2-manifolds},
       BookTitle = {In the tradition of Thurston III. Geometry and dynamics},
       ISBN = {978-3-031-43501-0; 978-3-031-43504-1; 978-3-031-43502-7},
       Pages = {105--167},
       Year = {2024},
       Publisher = {Cham: Springer},
       Language = {English},
       DOI = {10.1007/978-3-031-43502-7_5},
       Keywords = {57K20,57S05,54H11,20F65},
       zbMATH = {7918552}
   }
   

   The class of self-similar 2-manifolds consists of manifolds exhibiting a type of homogeneity akin to the 2-sphere and the Cantor set, and includes both the 2-sphere and the 2-sphere with a Cantor set removed. This chapter aims to provide a narrative thread between recent results on the structure of homeomorphism groups/mapping class groups of self-similar 2-manifolds, and also connections to classical structural results on the homeomorphism group of the 2-sphere and the Cantor set. In order to do this, we provide a survey of recent results, an exposition on classical results about homeomorphism groups, provide a treatment of the structure of stable sets, and prove extensions/strengthenings of the recent results surveyed. Of particular note, we establish the following theorems: (1) A characterization of homeomorphisms of (orientable) perfectly self-similar 2-manifolds that normally generate the group of (orientation-preserving) homeomorphisms -- a strengthening of a result of Malestein-Tao. (2) The homeomorphism group of a perfectly self-similar 2-manifold is strongly distorted -- an extension of a result of Calegari-Freedman for spheres. (3) The homeomorphism group of a perfectly tame 2-manifold is Steinhaus, and hence has the automatic continuity property -- an extension of a result of Mann in dimension two -- providing the first examples of homeomorphism groups of infinite-genus 2-manifolds with the Steinhaus property.
7. Bounded geometry with no bounded pants decomposition.

   Israel J. Math., 260 (2024), no. 1, 235--260.

   Joint with Ara Basmajian and Hugo Parlier

   abstract bibtex arXiv journal

   @Article{zbMATH07871045,
       Author = {Basmajian, Ara and Parlier, Hugo and Vlamis, Nicholas G.},
       Title = {Bounded geometry with no bounded pants decomposition},
       FJournal = {Israel Journal of Mathematics},
       Journal = {Isr. J. Math.},
       ISSN = {0021-2172},
       Volume = {260},
       Number = {1},
       Pages = {235--260},
       Year = {2024},
       Language = {English},
       DOI = {10.1007/s11856-023-2558-9},
       Keywords = {30F45},
       zbMATH = {7871045}
   }
   

   We construct a quasiconformally homogeneous hyperbolic Riemann surface-other than the hyperbolic plane-that does not admit a bounded pants decomposition. Also, given a connected orientable topological surface of infinite type with compact boundary components, we construct a complete hyperbolic metric on the surface that has bounded geometry but does not admit a bounded pants decomposition.
8. Mapping class groups with the Rokhlin property.

   Math. Z., 302 (2022), no. 3, 1343--1366.

   Joint with Justin Lanier

   abstract bibtex arXiv journal

   @Article{zbMATH07600570,
       Author = {Lanier, Justin and Vlamis, Nicholas G.},
       Title = {Mapping class groups with the {Rokhlin} property},
       FJournal = {Mathematische Zeitschrift},
       Journal = {Math. Z.},
       ISSN = {0025-5874},
       Volume = {302},
       Number = {3},
       Pages = {1343--1366},
       Year = {2022},
       Language = {English},
       DOI = {10.1007/s00209-022-03096-3},
       Keywords = {57K20,54H11,37B05,20E45},
       zbMATH = {7600570},
       Zbl = {1512.57023}
   }
   

   We classify the connected orientable 2-manifolds whose mapping class groups have a dense conjugacy class. We also show that the mapping class group of a connected orientable 2-manifold has a comeager conjugacy class if and only if the mapping class group is trivial.
9. Three perfect mapping class groups.

   New York J. Math., 27 (2021), 468--474.

   abstract bibtex arXiv journal

   @Article{zbMATH07319067,
       Author = {Vlamis, Nicholas G.},
       Title = {Three perfect mapping class groups},
       FJournal = {The New York Journal of Mathematics},
       Journal = {New York J. Math.},
       ISSN = {1076-9803},
       Volume = {27},
       Pages = {468--474},
       Year = {2021},
       Language = {English},
       Keywords = {57K20,57M07,22F99,20F99,37E30,57S05},
       URL = {nyjm.albany.edu/j/2021/27-18.html},
       zbMATH = {7319067},
       Zbl = {1459.57022}
   }
   

   We prove that the mapping class group of a surface obtained from removing a Cantor set from either the 2-sphere, the plane, or the interior of the closed 2-disk has no proper countable-index subgroups. The proof is an application of the automatic continuity of these groups, which was established by Mann. As corollaries, we see that these groups do not contain any proper finite-index subgroups and that each of these groups have trivial abelianization.
10. There are no exotic ladder surfaces.

    Ann. Fenn. Math., 47 (2022), no. 2, 1007--1023.

    Joint with Ara Basmajian

    abstract bibtex arXiv journal

    @Article{zbMATH07573918,
        Author = {Basmajian, Ara and Vlamis, Nicholas G.},
        Title = {There are no exotic ladder surfaces},
        FJournal = {Annales Fennici Mathematici},
        Journal = {Ann. Fenn. Math.},
        ISSN = {2737-114X},
        Volume = {47},
        Number = {2},
        Pages = {1007--1023},
        Year = {2022},
        Language = {English},
        Keywords = {30F99,30F45,30C62,57K20,57M10},
        zbMATH = {7573918},
        Zbl = {1527.30030}
    }
    

    It is an open problem to provide a characterization of quasiconformally homogeneous Riemann surfaces. We show that given the current literature, this problem can be broken into four open cases with respect to the topology of the underlying surface. The main result is a characterization in one of the these open cases; in particular, we prove that every quasiconformally homogeneous ladder surface is quasiconformally equivalent to a regular cover of a closed surface (or, in other words, there are no exotic ladder surfaces).
11. Isometry groups of infinite-genus hyperbolic surfaces.

    Math. Ann., 381 (2021), no. 1-2, 459--498.

    Joint with Tarik Aougab and Priyam Patel

    abstract bibtex arXiv journal

    @Article{zbMATH07412147,
        Author = {Aougab, Tarik and Patel, Priyam and Vlamis, Nicholas G.},
        Title = {Isometry groups of infinite-genus hyperbolic surfaces},
        FJournal = {Mathematische Annalen},
        Journal = {Math. Ann.},
        ISSN = {0025-5831},
        Volume = {381},
        Number = {1-2},
        Pages = {459--498},
        Year = {2021},
        Language = {English},
        DOI = {10.1007/s00208-021-02164-z},
        Keywords = {57M60,30F99},
        zbMATH = {7412147},
        Zbl = {1476.57041}
    }
    

    Given a 2-manifold, a fundamental question to ask is which groups can be realized as the isometry group of a Riemannan metric of constant curvature on the manifold. In this paper, we give a nearly complete classification of such groups for infinite-genus 2-manifolds with no planar ends. Surprisingly, we show there is an uncountable class of such 2-manifolds where every countable group can be realized as an isometry group (namely, those with self-similar end spaces). We apply this result to obtain obstructions to standard group theoretic properties for the groups of homeomorphisms, diffeomorphisms, and the mapping class groups of such 2-manifolds. For example, none of these groups satisfy the Tits Alternative; are coherent; are linear; are cyclically or linearly orderable; or are residually finite. As a second application, we give an algebraic rigidity result for mapping class groups.
12. Ends of non-metrizable manifolds: a generalized bagpipe theorem.

    Topology Appl., 310 (2022), Paper No. 108017, 30.

    Joint with David Fernández-Bretón

    abstract bibtex arXiv journal

    @Article{zbMATH07487231,
        Author = {Fern{\'a}ndez-Bret{\'o}n, David and Vlamis, Nicholas G.},
        Title = {Ends of non-metrizable manifolds: a generalized bagpipe theorem},
        FJournal = {Topology and its Applications},
        Journal = {Topology Appl.},
        ISSN = {0166-8641},
        Volume = {310},
        Pages = {30},
        Note = {Id/No 108017},
        Year = {2022},
        Language = {English},
        DOI = {10.1016/j.topol.2022.108017},
        Keywords = {57N99,54E35},
        zbMATH = {7487231},
        Zbl = {1502.57017}
    }
    

    We initiate the study of ends of non-metrizable manifolds and introduce the notion of short and long ends. Using the theory developed, we provide a characterization of (non-metrizable) surfaces that can be written as the topological sum of a metrizable manifold plus a countable number of "long pipes" in terms of their spaces of ends; this is a direct generalization of Nyikos's bagpipe theorem.
13. Big mapping class groups: an overview.

    In the tradition of Thurston—Geometry and Topology.

    Keni'chi Ohshika and Athanase Papadopoulos, eds., Springer, 2020, Chapter 12, 459--496.

    Joint with Javier Aramayona

    abstract bibtex arXiv journal

    @InCollection{zbMATH07444890,
        Author = {Aramayona, Javier and Vlamis, Nicholas G.},
        Title = {Big mapping class groups: an overview},
        BookTitle = {In the tradition of Thurston. Geometry and topology},
        ISBN = {978-3-030-55927-4; 978-3-030-55930-4; 978-3-030-55928-1},
        Pages = {459--496},
        Year = {2020},
        Publisher = {Cham: Springer},
        Language = {English},
        DOI = {10.1007/978-3-030-55928-1_12},
        Keywords = {57K20,57-02},
        zbMATH = {7444890},
        Zbl = {1479.57037}
    }
    

    We survey recent developments on mapping class groups of surfaces of infinite topological type.
14. Big mapping class groups acting on homology.

    Indiana Univ. Math. J., 70 (2021), no. 6, 2261--2294.

    Joint with Federica Fanoni and Sebastian Hensel

    abstract bibtex arXiv journal

    @Article{zbMATH07468279,
        Author = {Fanoni, Federica and Hensel, Sebastian and Vlamis, Nicholas G.},
        Title = {Big mapping class groups acting on homology},
        FJournal = {Indiana University Mathematics Journal},
        Journal = {Indiana Univ. Math. J.},
        ISSN = {0022-2518},
        Volume = {70},
        Number = {6},
        Pages = {2261--2294},
        Year = {2021},
        Language = {English},
        DOI = {10.1512/iumj.2021.70.8739},
        Keywords = {57K20},
        zbMATH = {7468279},
        Zbl = {1492.57008}
    }
    

    We study the action of (big) mapping class groups on the first homology of the corresponding surface. We give a precise characterization of the image of the induced homology representation.
15. Thurston norms of tunnel number-one manifolds.

    J. Knot Theory Ramifications, 28 (2019), no. 9, 1950056, 13.

    Joint with Natalia Pacheco-Tallaj and Kevin Schreve

    abstract bibtex arXiv journal

    @Article{zbMATH07103906,
        Author = {Pacheco-Tallaj, Natalia and Schreve, Kevin and Vlamis, Nicholas G.},
        Title = {Thurston norms of tunnel number-one manifolds},
        FJournal = {Journal of Knot Theory and its Ramifications},
        Journal = {J. Knot Theory Ramifications},
        ISSN = {0218-2165},
        Volume = {28},
        Number = {9},
        Pages = {13},
        Note = {Id/No 1950056},
        Year = {2019},
        Language = {English},
        DOI = {10.1142/S0218216519500561},
        Keywords = {57M27,57M05,20J05,20J06,57R19},
        zbMATH = {7103906},
        Zbl = {1425.57012}
    }
    

    The Thurston norm of a 3-manifold measures the complexity of surfaces representing two-dimensional homology classes. We study the possible unit balls of Thurston norms of 3-manifolds M with b1(M)=2, and whose fundamental groups admit presentations with two generators and one relator. We show that even among this special class, there are 3-manifolds such that the unit ball of the Thurston norm has arbitrarily many faces.
16. The first integral cohomology of pure mapping class groups.

    Int. Math. Res. Not. IMRN, (2020), no. 22, 8973--8996.

    Joint with Javier Aramayona and Priyam Patel

    abstract bibtex arXiv journal

    @Article{zbMATH07323408,
        Author = {Aramayona, Javier and Patel, Priyam and Vlamis, Nicholas G.},
        Title = {The first integral cohomology of pure mapping class groups},
        FJournal = {IMRN. International Mathematics Research Notices},
        Journal = {Int. Math. Res. Not.},
        ISSN = {1073-7928},
        Volume = {2020},
        Number = {22},
        Pages = {8973--8996},
        Year = {2020},
        Language = {English},
        DOI = {10.1093/imrn/rnaa229},
        Keywords = {57K20},
        zbMATH = {7323408},
        Zbl = {1462.57020}
    }
    

    It is a classical result of Powell that pure mapping class groups of connected, orientable surfaces of finite type and genus at least three are perfect. In stark contrast, we construct nontrivial homomorphisms from infinite-genus mapping class groups to the integers. Moreover, we compute the first integral cohomology group associated to the pure mapping class group of any connected orientable surface of genus at least 2 in terms of the surface's simplicial homology. In order to do this, we show that pure mapping class groups of infinite-genus surfaces split as a semi-direct product.
17. Algebraic and topological properties of big mapping class groups.

    Algebr. Geom. Topol., 18 (2018), no. 7, 4109--4142.

    Joint with Priyam Patel

    abstract bibtex arXiv journal

    @Article{zbMATH07006387,
        Author = {Patel, Priyam and Vlamis, Nicholas},
        Title = {Algebraic and topological properties of big mapping class groups},
        FJournal = {Algebraic \& Geometric Topology},
        Journal = {Algebr. Geom. Topol.},
        ISSN = {1472-2747},
        Volume = {18},
        Number = {7},
        Pages = {4109--4142},
        Year = {2018},
        Language = {English},
        DOI = {10.2140/agt.2018.18.4109},
        Keywords = {57K20,57M07,20E26,37E30,57S05},
        zbMATH = {7006387},
        Zbl = {1490.57016}
    }
    

    Let S be an orientable, connected surface with infinitely-generated fundamental group. The main theorem states that if the genus of S is finite and at least 4, then the isomorphism type of the pure mapping class group associated to S, denoted PMap(S), detects the homeomorphism type of S. As a corollary, every automorphism of PMap(S) is induced by a homeomorphism, which extends a theorem of Ivanov from the finite-type setting. In the process of proving these results, we show that PMap(S) is residually finite if and only if S has finite genus, demonstrating that the algebraic structure of PMap(S) can distinguish finite- and infinite-genus surfaces. As an independent result, we also show that Map(S) fails to be residually finite for any infinite-type surface S. In addition, we give a topological generating set for PMap(S) equipped with the compact-open topology. In particular, if S has at most one end accumulated by genus, then PMap(S) is topologically generated by Dehn twists, otherwise the Dehn twists along with handle shifts topologically generate.
18. Graphs of curves on infinite-type surfaces with mapping class group actions.

    Ann. Inst. Fourier (Grenoble), 68 (2018), no. 6, 2581--2612.

    Joint with Matthew Durham and Federica Fanoni

    abstract bibtex arXiv journal

    @Article{zbMATH07002326,
        Author = {Durham, Matthew Gentry and Fanoni, Federica and Vlamis, Nicholas G.},
        Title = {Graphs of curves on infinite-type surfaces with mapping class group actions},
        FJournal = {Annales de l'Institut Fourier},
        Journal = {Ann. Inst. Fourier},
        ISSN = {0373-0956},
        Volume = {68},
        Number = {6},
        Pages = {2581--2612},
        Year = {2018},
        Language = {English},
        DOI = {10.5802/aif.3217},
        Keywords = {57S05,37E30,20F65,57M07},
        zbMATH = {7002326},
        Zbl = {1416.57013}
    }
    

    We study when the mapping class group of an infinite-type surface S admits an action with unbounded orbits on a connected graph whose vertices are simple closed curves on S. We introduce a topological invariant for infinite-type surfaces that determines in many cases whether there is such an action. This allows us to conclude that, as non-locally compact topological groups, many big mapping class groups have nontrivial coarse geometry in the sense of Rosendal.
19. Coloring curves on surfaces.

    Forum Math. Sigma, 6 (2018), Paper No. e17, 42.

    Joint with Jonah Gaster and Joshua Greene

    abstract bibtex arXiv journal

    @Article{zbMATH06931988,
        Author = {Gaster, Jonah and Greene, Joshua Evan and Vlamis, Nicholas G.},
        Title = {Coloring curves on surfaces},
        FJournal = {Forum of Mathematics, Sigma},
        Journal = {Forum Math. Sigma},
        ISSN = {2050-5094},
        Volume = {6},
        Pages = {42},
        Note = {Id/No e17},
        Year = {2018},
        Language = {English},
        DOI = {10.1017/fms.2018.12},
        Keywords = {57M15,05C15},
        zbMATH = {6931988},
        Zbl = {1405.57008}
    }
    

    We study the chromatic number of the curve graph of a surface. We show that the chromatic number grows like k log k for the graph of separating curves on a surface of Euler characteristic -k. We also show that the graph of curves that represent a fixed non-zero homology class is uniquely t-colorable, where t denotes its clique number. Together, these results lead to the best known bounds on the chromatic number of the curve graph. We also study variations for arc graphs and obtain exact results for surfaces of low complexity. Our investigation leads to connections with Kneser graphs, the Johnson homomorphism, and hyperbolic geometry.
20. The Bridgeman-Kahn identity for hyperbolic manifolds with cusped boundary.

    Geom. Dedicata, 194 (2018), 81--97.

    Joint with Andrew Yarmola

    abstract bibtex arXiv journal

    @Article{zbMATH06894058,
        Author = {Vlamis, Nicholas G. and Yarmola, Andrew},
        Title = {The {Bridgeman}-{Kahn} identity for hyperbolic manifolds with cusped boundary},
        FJournal = {Geometriae Dedicata},
        Journal = {Geom. Dedicata},
        ISSN = {0046-5755},
        Volume = {194},
        Pages = {81--97},
        Year = {2018},
        Language = {English},
        DOI = {10.1007/s10711-017-0267-4},
        Keywords = {57M50,30F40,32Q45},
        zbMATH = {6894058},
        Zbl = {1395.57025}
    }
    

    In this note, we extend the Bridgeman-Kahn identity to all finite-volume orientable hyperbolic n-manifolds with totally geodesic boundary. In the compact case, Bridgeman and Kahn are able to express the manifold's volume as the sum of a function over only the orthospectrum. For manifolds with non-compact boundary, our extension adds terms corresponding to intrinsic invariants of boundary cusps.
21. Automorphisms of the compression body graph.

    J. Lond. Math. Soc. (2), 95 (2017), no. 1, 94--114.

    Joint with Ian Biringer

    abstract bibtex arXiv journal

    @Article{zbMATH06775070,
        Author = {Biringer, Ian and Vlamis, Nicholas G.},
        Title = {Automorphisms of the compression body graph},
        FJournal = {Journal of the London Mathematical Society. Second Series},
        Journal = {J. Lond. Math. Soc., II. Ser.},
        ISSN = {0024-6107},
        Volume = {95},
        Number = {1},
        Pages = {94--114},
        Year = {2017},
        Language = {English},
        DOI = {10.1112/jlms.12011},
        Keywords = {57M99},
        URL = {hdl.handle.net/2027.42/136257},
        zbMATH = {6775070},
        Zbl = {1380.57019}
    }
    

    When S is a closed, orientable surface with genus g(S)≥2, we show that the automorphism group of the compression body graph CB(S) is the mapping class group. Here, vertices are compression bodies with exterior boundary S, and edges connect pairs of compression bodies where one contains the other.
22. Basmajian's identity in higher Teichmüller-Thurston theory.

    J. Topol., 10 (2017), no. 3, 744--764.

    Joint with Andrew Yarmola

    abstract bibtex arXiv journal

    @Article{zbMATH06786978,
        Author = {Vlamis, Nicholas G. and Yarmola, Andrew},
        Title = {Basmajian's identity in higher {Teichm{\"u}ller}-{Thurston} theory},
        FJournal = {Journal of Topology},
        Journal = {J. Topol.},
        ISSN = {1753-8416},
        Volume = {10},
        Number = {3},
        Pages = {744--764},
        Year = {2017},
        Language = {English},
        DOI = {10.1112/topo.12022},
        Keywords = {32G15,30F60},
        zbMATH = {6786978},
        Zbl = {1376.32016}
    }
    

    We prove an extension of Basmajian's identity to n-Hitchin representations of compact bordered surfaces. For n=3, we show that this identity has a geometric interpretation for convex real projective structures analogous to Basmajian's original result. As part of our proof, we demonstrate that, with respect to the Lebesgue measure on the Frenet curve associated to a Hitchin representation, the limit set of an incompressible subsurface of a closed surface has measure zero. This generalizes a classical result in hyperbolic geometry. Finally, we recall the Labourie-McShane extension of the McShane-Mirzakhani identity to Hitchin representations and note a close connection to Basmajian's identity in both the hyperbolic and the Hitchin settings.
23. Quasiconformal homogeneity and subgroups of the mapping class group.

    Michigan Math. J., 64 (2015), no. 1, 53--75.

    abstract bibtex arXiv journal

    @Article{zbMATH06441170,
        Author = {Vlamis, Nicholas G.},
        Title = {Quasiconformal homogeneity and subgroups of the mapping class group},
        FJournal = {Michigan Mathematical Journal},
        Journal = {Mich. Math. J.},
        ISSN = {0026-2285},
        Volume = {64},
        Number = {1},
        Pages = {53--75},
        Year = {2015},
        Language = {English},
        DOI = {10.1307/mmj/1427203285},
        Keywords = {30C62,30F45},
        zbMATH = {6441170},
        Zbl = {1319.30012}
    }
    

    In the vein of Bonfert-Taylor, Bridgeman, Canary, and Taylor we introduce the notion of quasiconformal homogeneity for closed oriented hyperbolic surfaces restricted to subgroups of the mapping class group. We find uniform lower bounds for the associated quasiconformal homogeneity constants across all closed hyperbolic surfaces in several cases, including the Torelli group, congruence subgroups, and pure cyclic subgroups. Further, we introduce a counting argument providing a possible path to exploring a uniform lower bound for the nonrestricted quasiconformal homogeneity constant across all closed hyperbolic surfaces.
24. Moments of a length function on the boundary of a hyperbolic manifold.

    Algebr. Geom. Topol., 15 (2015), no. 4, 1909--1929.

    abstract bibtex arXiv journal

    @Article{zbMATH06486640,
        Author = {Vlamis, Nicholas G.},
        Title = {Moments of a length function on the boundary of a hyperbolic manifold},
        FJournal = {Algebraic \& Geometric Topology},
        Journal = {Algebr. Geom. Topol.},
        ISSN = {1472-2747},
        Volume = {15},
        Number = {4},
        Pages = {1909--1929},
        Year = {2015},
        Language = {English},
        DOI = {10.2140/agt.2015.15.1909},
        Keywords = {53D25,32Q45,30F40,51M10,57M50},
        zbMATH = {6486640},
        Zbl = {1334.53087}
    }
    

    In this paper we will study the statistics of the unit geodesic flow normal to the boundary of a hyperbolic manifold with non-empty totally geodesic boundary. Viewing the time it takes this flow to hit the boundary as a random variable, we derive a formula for its moments in terms of the orthospectrum. The first moment gives the average time for the normal flow acting on the boundary to again reach the boundary, which we connect to Bridgeman's identity (in the surface case), and the zeroth moment recovers Basmajian's identity. Furthermore, we are able to give explicit formulae for the first moment in the surface case as well as for manifolds of odd dimension. In dimension two, the summation terms are dilogarithms. In dimension three, we are able to find the moment generating function for this length function.