### Bobby Ramsey The Ohio State University Department of Mathematics 231 W 18th Avenue, MW 656 Columbus, OH 43210

#### To contact me...

 email address: ramsey.313@math.osu.edu office: MW 656

#### Papers and notes...

•  Extending properties to relatively hyperbolic groups With Daniel Ramras. Kyoto Journal of Mathematics ( to appear ) Consider a finitely generated group G that is relatively hyperbolic with respect to a family of subgroups $H_1$,...,$H_n$. We present an axiomatic approach to the problem of extending metric properties from the subgroups Hi to the full group $G$. We use this to show that both (weak) finite decomposition complexity and straight finite decomposition complexity are extendable properties. We also discuss the equivalence of two notions of straight finite decomposition complexity.

•  Strong embeddability and extensions of groups With Crichton Ogle and Ronghui Ji. We introduce the notion of strong embeddability for a metric space. This property lies between coarse embeddability and property A. A relative version of strong embeddability is developed in terms of a family of set maps on the metric space. When restricted to discrete groups, this yields relative coarse embeddability. We verify that groups acting on a metric space which is strongly embeddable has this relative strong embeddability, provided the stabilizer subgroups do. As a corollary, strong embeddability is preserved under group extensions. Relative property A groups are coarsely embeddable if their peripheral subgroups are.

•  Relative property A and relative amenability for countable groups With Crichton Ogle and Ronghui Ji. Advances in Mathematics 231 (2012), pp. 2734-2754. We define a relative property A for a countable group with respect to a finite family of subgroups. Many characterizations for relative property A are given. In particular a relative bounded cohomological characterization shows that if a group has property A relative to a family of subgroups and if each of those subgroups has property A, then the group has property A. This result leads to new classes of groups that have property A. In particular, groups are of property A if they act cocompactly on locally finite property A spaces of bounded geometry with stabilizers of property A. Specializing the definition of relative property A, an analogue definition of relative amenability for discrete groups are introduced and similar results are obtained.

•  On the Hochschild and cyclic (co)homology of rapid decay group algebras With Crichton Ogle and Ronghui Ji. J. Noncommut. Geom. 8 (2014), 45 -- 59. We show that the technical condition of solvable conjugacy bound, introduced in [JOR1], can be removed without affecting the main results of that paper. The result is a Burghelea-type description of the summands $HH_*^t(\mathcal{B}G)_{x}$ and $HC_*^t(\mathcal{B}G)_{x}$ for any bounding class $\mathcal{B}$, discrete group with word-length $(G,L)$ and conjugacy class $x \in G$. We use this description to prove the conjecture $\mathcal{B}$-SrBC of [JOR1] for a class of groups that goes well beyond the cases considered in that paper. In particular, we show that the conjecture $\ell^1$-SrBC (the Strong Bass Conjecture for the topological K-theory of $\ell^1(G)$) is true for all semihyperbolic groups which satisfy SrBC, a statement consistent with the rationalized Bost conjecture for such groups.

•  $\mathcal{B}$-Bounded Cohomology and Applications With Crichton Ogle and Ronghui Ji. International Journal of Algebra and Computation. Volume 23, Issue 1, February 2013, Pages 147-204. A discrete group with word-length $(G,L)$ is $\mathcal{B}$-isocohomological for a bounding classes $\mathcal{B}$ if the comparison map from $\mathcal{B}$-bounded cohomology to ordinary cohomology (with coefficients in $\mathbb{C}$) is an isomorphism; it is strongly $\mathcal{B}$-isocohomological if the same is true with arbitrary coefficients. In this paper we establish some basic conditions guaranteeing strong $\mathcal{B}$-isocohomologicality. In particular, we show strong $\mathcal{B}$-isocohomologicality for an $HF^{\infty}$ group $G$ is equivalent to the requirement that all of the weighted $G$-sensitive Dehn functions are $\mathcal{B}$-bounded. Such groups include all $\mathcal{B}$-asynchronously combable groups; moreover, the class of such groups is closed under constructions arising from groups acting on an acyclic complex. We also provide examples where the comparison map fails to be injective, as well as surjective, and give an example of a solvable group with quadratic first Dehn function, but exponential second Dehn function. Finally, a relative theory of $\mathcal{B}$-bounded cohomology of groups with respect to subgroups is introduced. Equivalent conditions on relative isocohomologicality is determined in terms of a new notion of relative Dehn functions and relative $HF^\infty$ property for groups with respect to subgroups. Applications for computing $\mathcal{B}$-bounded cohomology of groups are given in the context of relatively hyperbolic groups and developable complexes of groups.

•  Relatively Hyperbolic Groups, Rapid Decay Algebras, and a Generalization of the Bass Conjecture With Crichton Ogle and Ronghui Ji, and an Appendix by Crichton Ogle. Journal of Noncommutative Geometry. Volume 4, Issue 1, 2010, Pages 83-124 . By deploying dense subalgebras of $\ell^1(G)$ we generalize the Bass conjecture in terms of Connes’ cyclic homology theory. In particular, we propose a stronger version of the $\ell^1$-Bass Conjecture. We prove that hyperbolic groups relative to finitely many subgroups, each of which posses the polynomial conjugacy bound property and nilpotent periodicity property, satisfy the $\ell^1$-Stronger-Bass Conjecture. Moreover, we determine the conjugacy bound for relatively hyperbolic groups and compute the cyclic cohomology of the $\ell^1$-algebra of any discrete group.

•  The Isocohomological Property, Higher Dehn Functions, and Relatively Hyperbolic Groups With Ronghui Ji. Advances in Mathematics. Volume 222, Issue 1, 10 September 2009, Pages 255-280. We study the weak isocohomological property for finitely generated discrete groups. In the case the group has a classifying space the type of a simplicial complex with finitely many simplices in each dimension, ( called a type HF_\infty group), the property is related to the higher Dehn functions of this complex. In extending to the case of relatively hyperbolic groups we show that a group which is hyperbolic relative to combable groups is itself combable. In the case the subgroups are not combable, but merely HF_\infty, we construct a classifying space for the group, and bound the Dehn functions in terms of those of the subgroups.

•  A Spectral sequence for polynomially bounded cohomology Journal of Pure and Applied Algebra. Volume 217, Issue 6, June 2013, Pages 1153-1163. This is a writeup of the main results of my dissertation. We construct an analogue of the Lyndon-Hochschild-Serre spectral sequence in the context of polynomially bounded cohomology. For $G$ an extension of $Q$ by $H$, this spectral sequences converges to the polynomially bounded cohomology of $G$, $HP^*(G)$. If the extension is a polynomial extension in the sense of Noskov with $H$ and $Q$ isocohomological and $Q$ of type $HF^\infty$, the spectral sequence has $E_2^{p,q}$-term $HP^q(Q; HP^p(H))$, and $G$ is isocohomological for $\mathbb{C}$. By referencing results of Connes-Moscovici and Noskov if $H$ and $Q$ are both isocohomological and have the Rapid Decay property, then $G$ satisfies the Novikov conjecture.

•  A Generalization of the Lyndon-Hochschild-Serre Spectral Sequence for Polynomial Cohomology My Ph.D. Dissertation under the guidance of Ronghui Ji. I setup the framework for studying spectral sequences of bornological modules, then used this machinery to study the weak isocohomological property and its behavior under a certain class of extensions defined by Noskov. A reworked version has been submitted.

•  Cyclic Cohomology for Discrete Groups and its Applications With Ronghui Ji. 'Advances in Mathematics and its Applications', Y. Li, C.W. Shu, R. Ye, and K. Zuo eds, Univ. Sci. Tech. of China Press (2009).

•  Bounded Cohomology and Amenable Groups An elementary proof of a special case of a theorem of Johnson. The theorem says that for a discrete amenable group, the bounded cohomology vanishes in all positive degrees.

#### Talks...

•  September 2013, Wabash Modern Analysis Conference Exact families of maps and embedding relative property A groups Slides from the talk. Relative property A is a generalization of Yu's property A de ned for countable discrete groups. We will discuss how to adapt this to metric spaces by considering families of set maps. Using this characterization, we will show that if the peripheral subgroups are coarsely embeddable into Hilbert space, then so is the group.

•  September 2012, IUPUI Colloquium Relative property A for discrete groups Property A is a geometric condition on metric spaces introduced by Guoliang Yu for studying coarse embeddability into Hilbert spaces. It amounts to a nonequivariant generalization of amenability. Among many important consequences of property A are the validity of the Coarse Baum-Connes Conjecture (which, in turn, implies the Strong Novikov Conjecture), the Gromov-Lawson-Rosenberg Conjecture on the existence of metrics of positive scalar curvature, and Gromov’s zero-in-the-spectrum conjecture. In this talk, we generalize this by introducing relative property A for a countable discrete group with respect to a finite family of subgroups. Many characterizations for relative property A will be given, analogous to many different characterizations of property A. Using a recent cohomological characterization of Property A, found by Brodzki, Nowak, Niblo, and Wright, we are able to show that if a group has property A relative to a family of subgroups each of which has property A, then the group itself has property A. This result leads to new classes of groups that have property A.

•  March 2012, The Ohio State University, Topology Seminar Relative property A and relative amenability I discuss the various characterizations of amenability and property A for countable groups, paying particular interest to the cohomological versions. I then show how to generalize these to notions of 'relative property A' and 'relative amenability'.

•  June 2011, Heilbronn Institute, Geometric Group Theory Workshop The Bass Conjecture and Isocohomologicality.

•  April 2010, Purdue University, Operator Algebras Seminar The Isocohomological Property. Slides from the talk. In the first half of this talk, I describe the usual cyclic homology approach to the Bass conjecture through the nilpotency of the periodicity operator over each non-elliptic conjugacy class. I then introduce the isocohomological property and show how it can be exploited for verifying the $\ell^1$ Bass conjecture. I then discuss the relative isocohomological property for a finitely generated group with respect to a finite family of finitely generated subgroups, and give some examples.

•  March 2010, G3 Conference The Isocohomological Property. Slides from the talk. I setup and define the isocohomological property for a finitely generated group, and give a couple applications for motivation. I then give a sketch of the proof of the geometric characterization of the property in terms of weighted Dehn functions. I then discuss an example of Arzhantseva-Osin of an exponential growth solvable group with quadratic first Dehn function. By examining the behavior of the isocohomological property under nice types of extensions, we determine that the group is not polynomially isocohomological, but it is exponentially isocohomological. This gives the second weighted Dehn function as exponential, and bounds the second (unweighted) Dehn function between $e^n$ and $e^{(n^2)}$. (Osin in the audience verified that it really is $e^n$.) I finish by extending two results of Serre in group cohomology to the polynomially bounded framework: a long-exact sequence for groups acting on trees, and a spectral sequence for groups acting on a contractible complex.

•  February 2010, The Ohio State University, Geometric Group Theory Seminar The Isocohomological Property.

•  October 2009, Wabash Modern Analysis Conference Complexes of groups, the Isocohomological property, and Rapid Decay. I discuss the Rapid Decay property for finitely generated groups, and a proof of the RD property for Hyperbolic groups which can be extended to polynomial growth groups by considering `thickened triangles'. I then discuss the Isocohomological property and its geometric characterization through generalized Dehn function bounds. I conclude by discussing possible methods to verify these properties for groups acting on contractible complexes.

•  April 2009, Vanderbilt Topology and Group Theory Seminar The polynomially bounded conjugacy problem for relatively hyperbolic groups. Slides from the talk.

•  March 2009, Vanderbilt Noncommutative Geometry Seminar The isocohomological property, higher Dehn functions, and relatively hyperbolic groups. Slides from the talk.

•  February 2009, IUPUI Colloquium The isocohomological property, higher Dehn functions, and relatively hyperbolic groups. Slides from the talk.

•  September 2008, Wabash Modern Analysis Conference The isocohomological property, higher Dehn functions, and relatively hyperbolic groups.

•  November 2006, IUPUI Nonlinear PDE and Integrable Systems Seminar Geometric group theory, combability, and Dehn’s problems.

•  September 2005, Wabash Modern Analysis Conference Quasiconvex length functions and the Rapid Decay property.