University of Arkansas

Papers and Preprints (click for abstract and files or scroll down)

  1. Uniform fellow traveling between surgery paths in the sphere graph with Yulan Qing and Kasra Rafi
  2. Simultaneous construction of hyperbolic isometries with Caglar Uyanik
  3. ℓ²−homology of the free group
  4. ℓ²−torsion of free-by-cyclic groups
  5. When does a right-angled Artin group split over ℤ?
  6. An algorithm to detect full irreducibility by bounding the volume of periodic free factors with Johanna Mangahas and Alexandra Pettet
  7. Stable commuatator length in Baumslag−Solitar groups and quasimorphisms for tree actions with Max Forester and Joel Louwsma
  8. Abstract commensurators of right-angled Artin groups and mapping class groups with Christopher Leininger and Dan Margalit
  9. Uniform hyperbolicity of the curve graph via surgery sequences with Kasra Rafi and Saul Schleimer
  10. Whitehead graphs and separability in rank two with John Conant* and Nivetha Ramasubramanian*
  11. Relative twisting in Outer space with Alexandra Pettet
  12. Turn graphs and extremal surfaces in free groups with Noel Brady and Max Forester
  13. The geometry of right-angled Artin subgroups of mapping class groups with Christopher Leininger and Johanna Mangahas
  14. Current twisting and nonsingular matrices with Alexandra Pettet
  15. Twisting out fully irreducible automorphisms with Alexandra Pettet
  16. Morse theory and conjugacy classes of finite subgroups II with Noel Brady and Pallavi Dani
  17. Growth of intersection numbers for free group automorphismswith Jason Behrstock and Mladen Bestvina
  18. Morse theory and conjugacy classes of finite subgroups with Noel Brady and Pallavi Dani
  19. On the isomorphism problem for generalized Baumslag−Solitar groups with Max Forester
  20. Whitehead moves for G−trees with Max Forester
  21. Deformation spaces of G−trees and automorphisms of Baumslag−Solitar groups
  22. A fixed point theorem for deformation spaces of G−trees
  23. Contractibility of deformation spaces of G−trees
* undergraduate co-author

Matt Clay's articles on the arXiv

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  1. Uniform fellow traveling between surgery paths in the sphere graph with Yulan Qing and Kasra Rafi .pdf .ps Alg. and Geom. Topol. 17 (2017) 3751–3778.
    bound Abstract We show that the Hausdorff distance between any forward and any backward surgery paths in the sphere graph is at most 2. From this it follows that the Hausdorff distance between any two surgery paths with the same initial sphere system and same target sphere system is at most 4. Our proof relies on understanding how surgeries affect the Guirardel core associated to sphere systems. We show that applying a surgery is equivalent to performing a Rips move on the Guirardel core.
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  2. Simultaneous construction of hyperbolic isometries with Caglar Uyanik .pdf .ps
    bound Abstract. Given isometric actions by a group G on finitely many δ−hyperbolic metric spaces, we provide a sufficient condition that guarantees the existence of a single element in G that is hyperbolic for each action. As an application we prove a conjecture of Handel and Mosher regarding relatively fully irreducible subgroups and elements in the outer automorphism group of a free group.
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  3. ℓ²−homology of the free group .pdf .ps Expo. Math. 35 (2017) 133–148.
    bound Abstract. We analyze the reduced ℓ²−homology of a finitely generated nonabelian free group, 𝔽. Specifically, the projection map onto the space of harmonic ℓ²−1−chains is explicitly described and a weak isomorphism from (ℓ²(𝔽))rk(𝔽)−1 to the space of harmonic ℓ²−1−chains is given.
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  4. ℓ²−torsion of free-by-cyclic groups .pdf .ps Q. J. Math. 68 (2017) 617–634.
    bound Abstract. We provide an upper bound on the ℓ²−torsion of a free-by-cyclic group, −ρ(2)(𝔽 ⋊Φ ℤ), in terms of a relative train-track representative for Φ ∈ Aut(𝔽). Our result shares features with a theorem of Lück−Schick computing the ℓ²−torsion of the fundamental group of a 3−manifold that fibers over the circle in that it shows that the ℓ²−torsion is determined by the exponential dynamics of the monodromy. In light of the result of Lück−Schick, a special case of our bound is analogous to the bound on the volume of a 3−manifold that fibers over the circle with pseudo-Anosov monodromy by the normalized entropy recently demonstrated by Kojima−McShane.
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  5. When does a right-angled Artin group split over ℤ? .pdf .ps Internat. J. Algebra Comput. 24 (2014) 815−825.
    bound Abstract. We show that a right-angled Artin group, defined by a graph Γ that has at least three vertices, does not split over an infinite cyclic subgroup if and only if Γ is biconnected. Further, we compute JSJ−decompositions of 1−ended right-angled Artin groups over infinite cyclic subgroups.
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  6. An algorithm to detect full irreducibility by bounding the volume of periodic free factors with Johanna Mangahas and Alexandra Pettet .pdf .ps Michigan Math. J. 64 (2015) 279−292.
    bound Abstract. We provide an effective algorithm for determining whether an element φ of the outer automorphism group of a free group is fully irreducible. Our method produces a finite list which can be checked for periodic proper free factors.
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  7. Stable commuatator length in Baumslag−Solitar groups and quasimorphisms for tree actions with Max Forester and Joel Louwsma .pdf .ps Trans. Amer. Math. Soc. 368 (2016) 4751−4785.
    admissible surface Abstract. This paper has two parts, on Baumslag−Solitar groups and on general G−trees.
          In the first part we establish bounds for stable commutator length (scl) in Baumslag−Solitar groups. For a certain class of elements, we further show that scl is computable and takes rational values. We also determine exactly which of these elements admit extremal surfaces.
          In the second part we establish a universal lower bound of 1/12 for scl of suitable elements of any group acting on a tree. This is achieved by contructing efficient quasimorphisms. Calculations in the group BS(2, 3) show that this is the best possible universal bound, thus answering a question of Calegari and Fujiwara. We also establish scl bounds for acylindrical tree actions.
          Returning to Baumslag−Solitar groups, we show that their scl spectra have a uniform gap: no element has scl in the interval (0, 1/12).
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  8. Abstract commensurators of right-angled Artin groups and mapping class groups with Christopher Leininger and Dan Margalit .pdf .ps Math. Res. Lett. 21 (2014) 461−467.
    comm Abstract. We prove that, aside from the obvious exceptions, the mapping class group of a compact orientable surface is not abstractly commensurable with any right-angled Artin group. Our argument applies to various subgroups of the mapping class group−the subgroups generated by powers of Dehn twists and the terms of the Johnson filtration−and additionally to the outer automorphism group of a free group and to certain linear groups.
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  9. Uniform hyperbolicity of the curve graph via surgery sequences with Kasra Rafi and Saul Schleimer.pdf .ps Alg. and Geom. Topol. 14 (2014), 3325−3344.
    surgery Abstract. We prove that the curve graph C(1)(S) is Gromov-hyperbolic with a constant of hyperbolicity independent of the surface S. The proof is based on the proof of hyperbolicity of the free splitting complex by Handel and Mosher, as interpreted by Hilion and Horbez.
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  10. Whitehead graphs and separability in rank two with John Conant* and Nivetha Ramasubramanian*.pdf .ps Involve 7 (2014), 431−452.
    isSeparableFlowchart Abstract. By applying an algorithm of Stallings regarding separability of elements in a free group, we give an alternative approach to that of Osborne and Zieschang in describing all primitive elements in the free group of rank 2. As a result, we give a proof of a classical result of Nielsen, used by Osborne and Zieschang in their work, that the only automorphisms of F2 that act trivially on the abelianization are those defined by conjugation. Finally, we compute the probability that a Whitehead graph in rank 2 contains a cut vertex. We show that this probability is approximately 1/ℓ2 where ℓ is the number of edges in the graph.
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  11. Relative twisting in Outer space with Alexandra Pettet .pdf .ps J. Topol. Anal. 4 (2012), 173−201.
    twist Abstract. Subsurface projection has become indispensable in studying the geometry of the mapping class group and the curve complex of a surface. When the subsurface is an annulus, this projection is sometimes called relative twisting. We give two alternate versions of relative twisting for the outer automorphism group of a free group. We use this to describe sufficient conditions for when a folding path enters the thin part of Culler−Vogtmann's Outer space. As an application of our condition, we produce a sequence of fully irreducible outer automorphisms whose axes in Outer space travel through graphs with arbitrarily short cycles; we also describe the asymptotic behavior of their translation lengths.
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  12. Turn graphs and extremal surfaces in free groups with Noel Brady and Max Forester .pdf .ps Topology and Geometry in Dimension Three, 171−178, Contemp. Math., 560, 2011.
    S0-color Abstract. This notes provides an alternative account of Calegari's rationality theorem for stable commutator length in free groups.
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  13. The geometry of right angled Artin subgroups of mapping class groups with Christopher Leininger and Johanna Mangahas .pdf .ps Groups Geom. Dyn. 6 (2012), 249−278.
    raag Abstract. We describe sufficient conditions which guarantee that a finite set of mapping classes generate a right-angled Artin group quasi-isometrically embedded in the mapping class group. Moreover, under these conditions, the orbit map to Teichmüller space is a quasi-isometric embedding for both of the standard metrics. As a consequence, we produce infinitely many genus h surfaces (for any h at least 2) in the moduli space of genus g surfaces (for any g at least 3) for which the universal covers are quasi-isometrically embedded in the Teichmüller space.
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  14. Current twisting and nonsingular matrices with Alexandra Pettet .pdf .ps Comm. Math. Helv. 87 (2012), 385−407.
    IA-ses Abstract. We show for k ≥ 3, given any matrix in GL(k, ℤ) there is a hyperbolic fully irreducible automorphism of the free group of rank k whose induced action on ℤk is the given matrix.
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  15. Twisting out fully irreducible automorphisms with Alexandra Pettet .pdf .ps Geom. Funct. Anal. 20 (2010), 657−689.
    core Abstract. By a theorem of Thurston, in the subgroup of the mapping class group generated by Dehn twists around two curves which fill, every element not conjugate to a power of one of the twists is pseudo-Anosov. We prove an analogue of this theorem for the outer automorphism group of a free group.
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  16. Morse theory and conjugacy classes of finite subgroups II with Noel Brady and Pallavi Dani .pdf .ps Geom. Ded. 147 (2010), 1−14.
    fm Abstract. We construct a hyperbolic group containing a finitely presented subgroup, which has infinitely many conjugacy classes of finite-orider elements.
          We also use a version of Morse theory with high dimensional horizontal cells and use handle cancellation arguments to produce other examples of subgroups of CAT(0) groups with infinitely many conjugacy classes of finite-order elements.
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  17. Growth of intersection numbers for free group automorphisms with Jason Behrstock and Mladen Bestvina .pdf .ps J. of Topol. 3 (2010), 280−310.
    para-9 Abstract. For a fully irreducible automorphism φ of the free group Fk we compute the asymptotics of the intersection number ni(T,T'φn) for trees T, T' in Outer space. We also obtain qualitative information about the geometry of the Guirardel core for the trees T and T'φn for n large.
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  18. Morse theory and conjugacy classes of finite subgroups with Noel Brady and Pallavi Dani .pdf .ps Geom. Ded. 135 (2008), 15−22.
    mcg-example Abstract. We construct a CAT(0) group containing a finitely presented subgroup with infinitely many conjugacy classes of finite-order elements. Unlike previous examples (which were based on right-angled Artin groups) our ambient CAT(0) group does not contain any rank 3 free abelian subgroups.
          We also construct examples of type Fn inside mapping class groups, Aut(Fk), and Out(Fk) which have infinitely many conjugacy classes of finite-order elements.
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  19. On the isomorphism problem for generalized Baumslag-Solitar groups with Max Forester .pdf .ps Alg. and Geom. Topol. 8 (2008), 2289−2322.
    invariant Abstract. Generalized Baumslag-Solitar groups (GBS groups) are groups that act on trees with infinite cyclic edge and vertex stabilizers. Such an action is described by a labeled graph (essentially, the quotient graph of groups). This paper addresses the problem of determining whether two given labeled graphs define isomorphic groups; this is the isomorphism problem for GBS groups. There are two main results and some applications. First, we find necessary and sufficient conditions for a GBS group to be represented by only finitely many labeled graphs. These conditions can be checked effectively from any labeled graph. Then we show that the isomorphism problem is solvable for GBS groups whose labeled graphs have Betti number at most one.
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  20. Whitehead moves for G−trees with Max Forester .pdf .ps Bull. Lond. Math. Soc. 41 (2009), 205−212.
    amove Abstract. We generalize the familiar notion of a Whitehead move from Culler and Vogtmann's Outer space to the setting of deformation spaces of G−trees. Specifically, we show that there are two moves, each of which transforms a reduced G−tree into another reduced G−tree, that suffice to relate any two reduced trees in the same deformation space. There two moves further factor into three moves between reduced trees that have simples descriptions in terms of graphs of groups. This result has several applications.
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  21. Deformation spaces of G−trees and automorphisms of Baumslag−Solitar groups .pdf .ps Groups Geom. Dyn. 3 (2009), 39−69.
    sheet Abstract. We construct an invariant deformation retract of a deformation space of G-trees. We show that this complex is finite dimensional in certain cases and provide an example that is not finite dimensional. Using this complex we compute the automorphism group of the classical non-solvable Baumslag-Solitar groups BS(p,q). The most interesting case is when p properly divides q. Collins and Levin computed a presentation for Aut(BS(p,q)) in this case using algebraic methods. Our computation uses Bass-Serre theory to derive these presentations. Additionally, we provide a geometric argument showing Out(BS(p,q)) is not finitely-generated when p properly divides q.
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  22. A fixed point theorem for deformation spaces of G−trees .pdf .ps Comm. Math. Helv. 82 (2007), 237−246.
    realization Abstract. For a finitely generated free group Fn, of rank at least 2, any finite subgroup of Out(Fn) can be realized as a group of automorphisms of a graph with fundamental group Fn. This result is known as Out(Fn) Realization. This theorem is comparable to Nielsen Realization as proved by Kerchoff: for a closed surface with negative Euler characteristic, any finite subgroup of the mapping class group of the surface can be realized as a group of isometries of a hyperbolic surface. Both of these theorems have restatements in terms of fixed points. For nonnegative integers n we define a class of groups and prove a similar statement for their outer automorphism groups.
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  23. Contractibility of deformation spaces of G−trees .pdf .ps Alg. and Geom. Topol. 5 (2005), 1481−1503.
    thicken Abstract. Forester has defined spaces of simplicial tree actions for a finitely generated group, called deformation spaces. Culler and Vogtmann's Outer space is an example of a deformation space. Using ideas from Skora's proof of the contractibility of Outer space, we show that under some mild hypotheses deformation spaces are contractible.
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