We present the relation between stellar specific angular momentum $j_*$, stellar mass $M_*$, and bulge-to-total light ratio $beta$ for THINGS, CALIFA and Romanowsky & Fall datasets, exploring the existence of a fundamental plane between these paramet
ers as first suggested by Obreschkow & Glazebrook. Our best-fit $M_*-j_*$ relation yields a slope of $alpha = 1.03 pm 0.11$ with a trivariate fit including $beta$. When ignoring the effect of $beta$, the exponent $alpha = 0.56 pm 0.06$ is consistent with $alpha = 2/3$ predicted for dark matter halos. There is a linear $beta - j_*/M_*$ relation for $beta lesssim 0.4$, exhibiting a general trend of increasing $beta$ with decreasing $j_*/M_*$. Galaxies with $beta gtrsim 0.4$ have higher $j_*$ than predicted by the relation. Pseudobulge galaxies have preferentially lower $beta$ for a given $j_*/M_*$ than galaxies that contain classical bulges. Pseudobulge galaxies follow a well-defined track in $beta - j_*/M_*$ space, consistent with Obreschkow & Glazebrook, while galaxies with classical bulges do not. These results are consistent with the hypothesis that while growth in either bulge type is linked to a decrease in $j_*/M_*$, the mechanisms that build pseudobulges seem to be less efficient at increasing bulge mass per decrease in specific angular momentum than those that build classical bulges.
We prove a rigidity theorem that shows that, under many circumstances, quasi-isometric embeddings of equal rank, higher rank symmetric spaces are close to isometric embeddings. We also produce some surprising examples of quasi-isometric embeddings of
higher rank symmetric spaces. In particular, we produce embeddings of $SL(n,mathbb R)$ into $Sp(2(n-1),mathbb R)$ when no isometric embeddings exist. A key ingredient in our proofs of rigidity results is a direct generalization of the Mostow-Morse Lemma in higher rank. Typically this lemma is replaced by the quasi-flat theorem which says that maximal quasi-flat is within bounded distance of a finite union of flats. We improve this by showing that the quasi-flat is in fact flat off of a subset of codimension $2$.
In this paper, which is the continuation of [EFW2], we complete the proof of the quasi-isometric rigidity of Sol and the lamplighter groups. The results were announced in [EFW1].
In this paper, we prove that certain spaces are not quasi-isometric to Cayley graphs of finitely generated groups. In particular, we answer a question of Woess and prove a conjecture of Diestel and Leader by showing that certain homogeneous graphs ar
e not quasi-isometric to a Cayley graph of a finitely generated group. This paper is the first in a sequence of papers proving results announced in [EFW0]. In particular, this paper contains many steps in the proofs of quasi-isometric rigidity of lattices in Sol and of the quasi-isometry classification of lamplighter groups. The proofs of those results are completed in [EFW1]. The method used here is based on the idea of coarse differentiation introduced in [EFW0].
In this note, we announce the first results on quasi-isometric rigidity of non-nilpotent polycyclic groups. In particular, we prove that any group quasi-isometric to the three dimenionsional solvable Lie group Sol is virtually a lattice in Sol. We pr
ove analogous results for groups quasi-isometric to $R ltimes R^n$ where the semidirect product is defined by a diagonalizable matrix of determinant one with no eigenvalues on the unit circle. Our approach to these problems is to first classify all self quasi-isometries of the solvable Lie group. Our classification of self quasi-isometries for $R ltimes R^n$ proves a conjecture made by Farb and Mosher in [FM4]. Our techniques for studying quasi-isometries extend to some other classes of groups and spaces. In particular, we characterize groups quasi-isometric to any lamplighter group, answering a question of de la Harpe [dlH]. Also, we prove that certain Diestel-Leader graphs are not quasi-isometric to any finitely generated group, verifying a conjecture of Diestel and Leader from [DL] and answering a question of Woess from [SW],[Wo1]. We also prove that certain non-unimodular, non-hyperbolic solvable Lie groups are not quasi-isometric to finitely generated groups. The results in this paper are contributions to Gromovs program for classifying finitely generated groups up to quasi-isometry [Gr2]. We introduce a new technique for studying quasi-isometries, which we refer to as coarse differentiation.
Let G be a subgroup of finite index in SL(n,Z) for N > 4. Suppose G acts continuously on a manifold M, with fundamental group Z^n, preserving a measure that is positive on open sets. Further assume that the induced G action on H^1(M) is non-trivial.
We show there exists a finite index subgroup G of G and a G equivariant continuous map from M to the n-torus that induces an isomorphism on fundamental groups. We prove more general results providing continuous quotients in cases where the fundamental group of M surjects onto a finitely generated torsion free nilpotent group. We also give some new examples of manifolds with G actions to which the theorems apply.
