Introduction 1. Graphs of groups of finite index; unimodularity Appendix A: Nonabelian cohomology of a graph 2. Finite groupings of edge indexed graphs; volumes 3. Automorphism groups of locally finite trees; unimodularity 4. Existence, conjugacy, and commensurability of uniform lattices 5. Volumes, Euler characteristics, and ranks 6. Finiteness properties Appendix B: Commensurators 7. Nonfiniteness phenomena References O. INTRODUCTION Let X be a locally finite tree. Then G = Aut(X) is a locally compact group; the stabilizers G x are open and profinite. A subgroup r:5 G is discrete if all rx are finite. We then call r a lattice if Vol(I\ \X) = L Tfl xEf\X xis finite, and a uniform lattice if the graph I\X is finite. These are the objects of our study. It is fruitful to think of (G, X, r) as a combinatorial analogue of (PSL 2 (R), upper half-plane, Fuchsian group). An even more direct analogy

A branched covering M-> N of degree d between closed surfaces determines a collection L'D of partitions of d-its "branch data"-corresponding to the set of branch points. The collection of partitions must satisfy certain obvious conditions implied by the Riemann-Hurwitz formula. This paper investigates the extent to which any such finite collection 6D of partitions of d can be realized as the branch data of a suitable branched covering. If N is not the 2-sphere, such data can always be realized. If ''D contains sufficiently many elements compared to d, then it can be realized. And whenever d is nonprime, examples are constructed of nonrealizable data.

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The object of this paper is to study the various curvatures which appear in differential geometry in different contexts. We shall point out the common analytic features underlying the tensors which appear in the definitions of these curvatures and equally emphasize the curvature functions themselves. Two basic examples are:1. The second fundamental form of a hypersurface. The corresponding curvature function is the socalled normal curvature.2. The Riemann curvature tensor of a Riemann manifold. The corresponding curvature function is sectional curvature.In [5-1 and [8] we have proved that under certain conditions a diffeomorphism of hypersurfaces preserving normal curvature is a congruence and a diffeomorphism of Riemann manifolds preserving sectional curvature is an isometry. Motivation of the present work was to seek generalization of these results to Gauss Kronecker curvatures. These curvatures arise as follows: the Gaussian definition of sectional curvature of a 2-dimensional surface via the Gauss map into the unit sphere can be imitated for a hypersurface in Euclidean space of arbitrary dimension. If the dimension of the hypersurface is even then the function obtained this way depends only on the intrinsic geometry of the hypersurface, and more generally for an arbitrary Riemann manifold (M, g) for each even positive integer 2q one can define a curvature function Kq which is a function on the Grassmann bundle of 2q-planes on the manifold. K 1 coincides with the sectional curvature. We shall call Kq = the q-th Gauss Kronecker curvature of M. It will be shown that /f dim M > 4q then generically a Kq-preserving diffeomorphism is an isometry (cf. Theorem 8.2).However this result is not as strong as in the case of sectional curvature. We have been able to carry through the algebraic part completely and in the process obtained some higher order presumably new conformal invariants. However the differential identities that we derive do not suffice to prove as strong a result as the one for sectional curvature.In [5-1 the result for sectional curvature preserving maps is proved completely only for dimension > 4. The case of 3-dimensional manifolds was largely unsettled. In this paper the proof of the result is greatly simplified and also two important cases when the dimension equals 3 are 13 Math. Ann. 199

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Donor age is one of the major concerns in bone marrow transplantation, as the aged hematopoietic stem cells (HSCs) fail to engraft efficiently. Here, using murine system, we show that a brief interaction of aged HSCs with young mesenchymal stromal cells (MSCs) rejuvenates them and restores their functionality via inter-cellular transfer of microvesicles (MVs) containing autophagy-related mRNAs. Importantly, we show that MSCs gain activated AKT signaling as a function of aging. Activated AKT reduces the levels of autophagy-related mRNAs in their MVs, and partitions miR-17 and miR-34a into their exosomes, which upon transfer into HSCs downregulate their autophagy-inducing mRNAs. Our data identify previously unknown mechanisms operative in the niche-mediated aging of HSCs. Inhibition of AKT in aged MSCs increases the levels of autophagy-related mRNAs in their MVs and reduces the levels of miR-17 and miR-34a in their exosomes. Interestingly, transplantation experiments showed that the rejuvenating power of these "rescued" MVs is even better than that of the young MVs. We demonstrate that such ex vivo rejuvenation of aged HSCs could expand donor cohort and improve transplantation efficacy. Stem Cells 2018;36:420-433.

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