Packing subgroups in relatively hyperbolic groups

We introduce the bounded packing property for a subgroup of a countable discrete group G. This property gives a finite upper bound on the number of left cosets of the subgroup that are pairwise close in G. We establish basic properties of bounded packing, and give many examples; for instance, every subgroup of a countable, virtually nilpotent group has bounded packing. We explain several natural connections between bounded packing and group actions on CAT(0) cube complexes. Our main result establishes the bounded packing of relatively quasiconvex subgroups of a relatively hyperbolic group, under mild hypotheses. As an application, we prove that relatively quasiconvex subgroups have finite height and width, properties that strongly restrict the way families of distinct conjugates of the subgroup can intersect. We prove that an infinite, nonparabolic relatively quasiconvex subgroup of a relatively hyperbolic group has finite index in its commensurator. We also prove a virtual malnormality theorem for separable, relatively quasiconvex subgroups, which is new even in the word hyperbolic case.Comment: 45 pages, 2 figures. To appear in Geom. Topol. v2: Updated to address concerns of the referee. Added theorem that an infinite, nonparabolic relatively quasiconvex subgroup H of a relatively hyperbolic group has finite index in its commensurator. Added several new geometric results to Section 7. Theorem 8.9 on packing relative to peripheral subgroups is ne

Synaptic nanomodules underlie the organization and plasticity of spine synapses.

Experience results in long-lasting changes in dendritic spine size, yet how the molecular architecture of the synapse responds to plasticity remains poorly understood. Here a combined approach of multicolor stimulated emission depletion microscopy (STED) and confocal imaging in rat and mouse demonstrates that structural plasticity is linked to the addition of unitary synaptic nanomodules to spines. Spine synapses in vivo and in vitro contain discrete and aligned subdiffraction modules of pre- and postsynaptic proteins whose number scales linearly with spine size. Live-cell time-lapse super-resolution imaging reveals that NMDA receptor-dependent increases in spine size are accompanied both by enhanced mobility of pre- and postsynaptic modules that remain aligned with each other and by a coordinated increase in the number of nanomodules. These findings suggest a simplified model for experience-dependent structural plasticity relying on an unexpectedly modular nanomolecular architecture of synaptic proteins

Finiteness properties of cubulated groups

We give a generalized and self-contained account of Haglund-Paulin's wallspaces and Sageev's construction of the CAT(0) cube complex dual to a wallspace. We examine criteria on a wallspace leading to finiteness properties of its dual cube complex. Our discussion is aimed at readers wishing to apply these methods to produce actions of groups on cube complexes and understand their nature. We develop the wallspace ideas in a level of generality that facilitates their application. Our main result describes the structure of dual cube complexes arising from relatively hyperbolic groups. Let H_1,...,H_s be relatively quasiconvex codimension-1 subgroups of a group G that is hyperbolic relative to P_1,...,P_r. We prove that G acts relatively cocompactly on the associated dual CAT(0) cube complex C. This generalizes Sageev's result that C is cocompact when G is hyperbolic. When P_1,...,P_r are abelian, we show that the dual CAT(0) cube complex C has a G-cocompact CAT(0) truncation.Comment: 58 pages, 12 figures. Version 3: Revisions and slightly improved results in Sections 7 and 8. Several theorem numbers have changed from the previous versio

Local Quasiconvexity of Groups acting on Small Cancellation Complexes

Given a group acting cellularly and cocompactly on a simply-connected 2-complex, we provide a criterion establishing that all finitely generated subgroups have quasiconvex orbits. This work generalizes the "perimeter method". As an application, we show that high-powered one-relator products A \ast B / \nclose{r^n} are coherent if $A$ and $B$ are coherent.Comment: version 1. 14 pages, 4 figure

Sotatercept safety and effects on hemoglobin, bone, and vascular calcification

Introduction: Patients with end-stage kidney disease (ESKD) exhibit anemia, chronic kidney disease‒mineral bone disorder (CKD-MBD), and cardiovascular disease. The REN-001 and REN-002 phase II, multicenter, randomized studies examined safety, tolerability, and effects of sotatercept, an ActRIIA-IgG1 fusion protein trap, on hemoglobin concentration; REN-001 also explored effects on bone mineral density (BMD) and abdominal aortic vascular calcification. Methods: Forty-three patients were treated in REN-001 (dose range: sotatercept 0.3‒0.7 mg/kg or placebo subcutaneously [s.c.] for 200 days) and 50 in REN-002 (dose range: 0.1‒0.4 mg/kg i.v. and 0.13‒0.5 mg/kg s.c. for 99 days). Results: In REN-001, frequency of achieving target hemoglobin response (\u3e10 g/dl [6.21 mmol/l]) with sotatercept was dose-related and greater than placebo (0.3 mg/kg: 33.3%; 0.5 mg/kg: 62.5%; 0.7 mg/kg: 77.8%; 0.7 mg/kg [doses 1 and 2]/0.4 mg/kg [doses 3‒15]: 33.3%; placebo: 27.3%). REN-002 hemoglobin findings were similar (i.v.: 16.7%-57.1%; s.c.: 11.1%‒42.9%). Dose-related achievement of ≥2% increase in femoral neck cortical BMD was seen among only REN-001 patients receiving sotatercept (0.3‒0.7 mg/kg: 20.0%‒57.1%; placebo: 0.0%). Abdominal aortic vascular calcification was slowed in a dose-related manner, with a ≤15% increase in Agatston score achieved by more REN-001 sotatercept versus placebo patients (60%‒100% vs. 16.7%). The most common adverse events during treatment were hypertension, muscle spasm, headache, arteriovenous fistula site complication, and influenza observed in both treatment and placebo groups. Conclusion: In patients with ESKD, sotatercept exhibited a favorable safety profile and was associated with trends in dose-related slowing of vascular calcification. Less-consistent trends in improved hemoglobin concentration and BMD were observed

Potent spinal parenchymal AAV9-mediated gene delivery by subpial injection in adult rats and pigs.

Effective in vivo use of adeno-associated virus (AAV)-based vectors to achieve gene-specific silencing or upregulation in the central nervous system has been limited by the inability to provide more than limited deep parenchymal expression in adult animals using delivery routes with the most clinical relevance (intravenous or intrathecal). Here, we demonstrate that the spinal pia membrane represents the primary barrier limiting effective AAV9 penetration into the spinal parenchyma after intrathecal AAV9 delivery. We develop a novel subpial AAV9 delivery technique and AAV9-dextran formulation. We use these in adult rats and pigs to show (i) potent spinal parenchymal transgene expression in white and gray matter including neurons, glial and endothelial cells after single bolus subpial AAV9 delivery; (ii) delivery to almost all apparent descending motor axons throughout the length of the spinal cord after cervical or thoracic subpial AAV9 injection; (iii) potent retrograde transgene expression in brain motor centers (motor cortex and brain stem); and (iv) the relative safety of this approach by defining normal neurological function for up to 6 months after AAV9 delivery. Thus, subpial delivery of AAV9 enables gene-based therapies with a wide range of potential experimental and clinical utilizations in adult animals and human patients

Ephrin-B3 controls excitatory synapse density through cell-cell competition for EphBs

Cortical networks are characterized by sparse connectivity, with synapses found at only a subset of axo-dendritic contacts. Yet within these networks, neurons can exhibit high connection probabilities, suggesting that cell-intrinsic factors, not proximity, determine connectivity. Here, we identify ephrin-B3 (eB3) as a factor that determines synapse density by mediating a cell-cell competition that requires ephrin-B-EphB signaling. In a microisland culture system designed to isolate cell-cell competition, we find that eB3 determines winning and losing neurons in a contest for synapses. In a Mosaic Analysis with Double Markers (MADM) genetic mouse model system in vivo the relative levels of eB3 control spine density in layer 5 and 6 neurons. MADM cortical neurons in vitro reveal that eB3 controls synapse density independently of action potential-driven activity. Our findings illustrate a new class of competitive mechanism mediated by trans-synaptic organizing proteins which control the number of synapses neurons receive relative to neighboring neurons

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