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Chris Connell

Publications and source records attributed to Chris Connell.

25 records · Page 2Linked to original sources
arXiv2016

A vanishing theorem for the homology of discrete subgroups of $\mathrm{Sp}(n,1)$ and $\mathrm{F}_4^{-20}$

For any discrete, torsion-free subgroup $Γ$ of $\mathrm{Sp}(n,1)$ (resp.\ $\mathrm{F}_4^{-20}$) with no parabolic elements, we prove that $H_{4n-1}(Γ;V)=0$ (resp.\ $H_i(Γ;V)=0$ for $i=13,14,15$) for any $Γ$--module $V$. The main technical advance is a new bound on the $p$--Jacobian of the barycenter map of Besson--Courtois--Gallot. We also apply this estimate to obtain an inequality between the critical exponent and homological dimension of $Γ$, improving on work of M.~Kapovich.

math.GT↗
arXiv2015

Harmonic and invariant measures on foliated spaces

We consider the family of harmonic measures on a lamination $\mathcal{L}$ of a compact space $X$ by locally symmetric spaces $L$ of noncompact type, i.e. $L\simeq Γ_L\backslash G/K$. We establish a natural bijection between these measures and the measures on an associated lamination foliated by $G$-orbits, $\hat{\mathcal{L}}$ which are right invariant under a minimal parabolic (Borel) subgroup $B < G$. In the special case when $G$ is split, these measures correspond to the measures that are invariant under both the Weyl chamber flow and the stable horospherical flows on a certain bundle over the associated Weyl chamber lamination. We also show that the measures on $\hat{\mathcal{L}}$ right invariant under two distinct minimal parabolics, and therefore all of $G$, are in bijective correspondence with the holonomy-invariant ones.

math.DS↗
arXiv2007

Smooth Volume Rigidity for Manifolds with Negatively Curved Targets

We establish conditions for a continuous map of nonzero degree between a smooth closed manifold and a negatively curved manifold of dimension greater than four to be homotopic to a smooth cover, and in particular a diffeomorphism when the degree is one. The conditions hold when the volumes or entropy-volumes of the two manifolds differ by less than a uniform constant after an appropriate normalization of the metrics. The results are qualitatively sharp in the sense that the dependencies are necessary. We give a number of corollaries.

math.DG↗
arXiv2005

Harmonicity of Gibbs measures

In this paper we extend the construction of random walks with a prescribed Poisson boundary to the case of measures in the class of a generalized Gibbs state. The price for dropping the $α$-quasiconformal assumptions is that we must restrict our attention to CAT($-κ$) groups. Apart from the new estimates required, we prove a new approximation scheme to provide a positive basis for positive functions in a metric measure space.

math.GR↗
arXiv2004

Harmonicity of quasiconformal measures and Poisson boundaries of hyperbolic spaces

We consider a group G of isometries acting on a (not necessarily geodesic) delta-hyperbolic space X and possessing a radial limit set of full measure within its limit set. For any continuous quasiconformal measure w supported on the limit set, we produce a stationary measure m on G. Moreover the limit set together with w forms a m-boundary and w is harmonic with respect to the random walk induced by m. In the case when X is a CAT(-1) space and G acts cocompactly, for instance, we show that m has finite first moment. This implies that the boundary of X with w is the unique Poisson boundary for m. As a bi-product, we establish sufficient conditions for a set of continuous functions to form a positive basis, either in the L^1 or sup norm, for the space of uniformly positive lower-semicontinuous functions on a general metric measure space.

math.GR↗