Searching for Hyperbolicity

This is an expository paper, based on by a talk given at the AWM Research Symposium 2017. It is intended as a gentle introduction to geometric group theory with a focus on the notion of hyperbolicity, a theme that has inspired the field from its inception to current-day research

On the density of periodic configurations in strongly irreducible subshifts

Let $G$ be a residually finite group and let $A$ be a finite set. We prove that if $X \subset A^G$ is a strongly irreducible subshift of finite type containing a periodic configuration then periodic configurations are dense in $X$. The density of periodic configurations implies in particular that every injective endomorphism of $X$ is surjective and that the group of automorphisms of $X$ is residually finite. We also introduce a class of subshifts $X \subset A^\Z$, including all strongly irreducible subshifts and all irreducible sofic subshifts, in which periodic configurations are dense

Accidental parabolics and relatively hyperbolic groups

By constructing, in the relative case, objects analoguous to Rips and Sela's canonical representatives, we prove that the set of images by morphisms without accidental parabolic, of a finitely presented group in a relatively hyperbolic group, is finite, up to conjugacy.Comment: Revision, 24 pages, 4 figure

Existential questions in (relatively) hyperbolic groups {\it and} Finding relative hyperbolic structures

This arXived paper has two independant parts, that are improved and corrected versions of different parts of a single paper once named "On equations in relatively hyperbolic groups". The first part is entitled "Existential questions in (relatively) hyperbolic groups". We study there the existential theory of torsion free hyperbolic and relatively hyperbolic groups, in particular those with virtually abelian parabolic subgroups. We show that the satisfiability of systems of equations and inequations is decidable in these groups. In the second part, called "Finding relative hyperbolic structures", we provide a general algorithm that recognizes the class of groups that are hyperbolic relative to abelian subgroups.Comment: Two independant parts 23p + 9p, revised. To appear separately in Israel J. Math, and Bull. London Math. Soc. respectivel

A Combination Theorem for Metric Bundles

We define metric bundles/metric graph bundles which provide a purely topological/coarse-geometric generalization of the notion of trees of metric spaces a la Bestvina-Feighn in the special case that the inclusions of the edge spaces into the vertex spaces are uniform coarsely surjective quasi-isometries. We prove the existence of quasi-isometric sections in this generality. Then we prove a combination theorem for metric (graph) bundles (including exact sequences of groups) that establishes sufficient conditions, particularly flaring, under which the metric bundles are hyperbolic. We use this to give examples of surface bundles over hyperbolic disks, whose universal cover is Gromov-hyperbolic. We also show that in typical situations, flaring is also a necessary condition.Comment: v3: Major revision: 56 pages 5 figures. Many details added. Characterization of convex cocompact subgroups of mapping class groups of surfaces with punctures in terms of relative hyperbolicity given v4: Final version incorporating referee comments: 63 pages 5 figures. To appear in Geom. Funct. Ana

Effective-Range Expansion of the Neutron-Deuteron Scattering Studied by a Quark-Model Nonlocal Gaussian Potential

The S-wave effective range parameters of the neutron-deuteron (nd) scattering are derived in the Faddeev formalism, using a nonlocal Gaussian potential based on the quark-model baryon-baryon interaction fss2. The spin-doublet low-energy eigenphase shift is sufficiently attractive to reproduce predictions by the AV18 plus Urbana three-nucleon force, yielding the observed value of the doublet scattering length and the correct differential cross sections below the deuteron breakup threshold. This conclusion is consistent with the previous result for the triton binding energy, which is nearly reproduced by fss2 without reinforcing it with the three-nucleon force.Comment: 21 pages, 6 figures and 6 tables, submitted to Prog. Theor. Phy

NKT sublineage specification and survival requires the ubiquitin-modifying enzyme TNF AIP3/A20

Natural killer T (NKT) cells are innate lymphocytes that differentiate into NKT1, NKT2, and NKT17 sublineages during development. However, the signaling events that control NKT sublineage specification and differentiation remain poorly understood. Here, we demonstrate that the ubiquitin-modifying enzyme TNF AIP3/A20, an upstream regulator of T cell receptor (TCR) signaling in T cells, is an essential cell-intrinsic regulator of NKT differentiation. A20 is differentially expressed during NKT cell development, regulates NKT cell maturation, and specifically controls the differentiation and survival of NKT1 and NKT2, but not NKT17, sublineages. Remaining A20-deficient NKT1 and NKT2 thymocytes are hyperactivated in vivo and secrete elevated levels of Th1 and Th2 cytokines after TCR ligation in vitro. Defective NKT development was restored by compound deficiency of MALT1, a key downstream component of TCR signaling in T cells. These findings therefore show that negative regulation of TCR signaling during NKT development controls the differentiation and survival of NKT1 and NKT2 cells

Selective C-Rel Activation via Malt1 Controls Anti-Fungal TH-17 Immunity by Dectin-1 and Dectin-2

C-type lectins dectin-1 and dectin-2 on dendritic cells elicit protective immunity against fungal infections through induction of TH1 and TH-17 cellular responses. Fungal recognition by dectin-1 on human dendritic cells engages the CARD9-Bcl10-Malt1 module to activate NF-κB. Here we demonstrate that Malt1 recruitment is pivotal to TH-17 immunity by selective activation of NF-κB subunit c-Rel, which induces expression of TH-17-polarizing cytokines IL-1β and IL-23p19. Malt1 inhibition abrogates c-Rel activation and TH-17 immunity to Candida species. We found that Malt1-mediated activation of c-Rel is similarly essential to induction of TH-17-polarizing cytokines by dectin-2. Whereas dectin-1 activates all NF-κB subunits, dectin-2 selectively activates c-Rel, signifying a specialized TH-17-enhancing function for dectin-2 in anti-fungal immunity by human dendritic cells. Thus, dectin-1 and dectin-2 control adaptive TH-17 immunity to fungi via Malt1-dependent activation of c-Rel