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Search: MSC category 20F65 ( Geometric group theory [See also 05C25, 20E08, 57Mxx] )

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1. CMB 2016 (vol 60 pp. 54)

Button, Jack
Tubular Free by Cyclic Groups Act Freely on CAT(0) Cube Complexes
We identify when a tubular group (the fundamental group of a finite graph of groups with $\mathbb{Z}^2$ vertex and $\mathbb{Z}$ edge groups) is free by cyclic and show, using Wise's equitable sets criterion, that every tubular free by cyclic group acts freely on a CAT(0) cube complex.

Keywords:CAT(0), tubular group
Categories:20F65, 20F67, 20E08

2. CMB Online first

Louder, Larsen; Wilton, Henry
Stackings and the $W$-cycles conjecture
We prove Wise's $W$-cycles conjecture: Consider a compact graph $\Gamma'$ immersing into another graph $\Gamma$. For any immersed cycle $\Lambda:S^1\to \Gamma$, we consider the map $\Lambda'$ from the circular components $\mathbb{S}$ of the pullback to $\Gamma'$. Unless $\Lambda'$ is reducible, the degree of the covering map $\mathbb{S}\to S^1$ is bounded above by minus the Euler characteristic of $\Gamma'$. As a corollary, any finitely generated subgroup of a one-relator group has finitely generated Schur multiplier.

Keywords:free groups, one-relator groups, right-orderability

3. CMB 2016 (vol 60 pp. 77)

Christ, Michael; Rieffel, Marc A.
Nilpotent Group C*-algebras as Compact Quantum Metric Spaces
Let $\mathbb{L}$ be a length function on a group $G$, and let $M_\mathbb{L}$ denote the operator of pointwise multiplication by $\mathbb{L}$ on $\lt(G)$. Following Connes, $M_\mathbb{L}$ can be used as a ``Dirac'' operator for the reduced group C*-algebra $C_r^*(G)$. It defines a Lipschitz seminorm on $C_r^*(G)$, which defines a metric on the state space of $C_r^*(G)$. We show that for any length function satisfying a strong form of polynomial growth on a discrete group, the topology from this metric coincides with the weak-$*$ topology (a key property for the definition of a ``compact quantum metric space''). In particular, this holds for all word-length functions on finitely generated nilpotent-by-finite groups.

Keywords:group C*-algebra, Dirac operator, quantum metric space, discrete nilpotent group, polynomial growth
Categories:46L87, 20F65, 22D15, 53C23, 58B34

4. CMB 2010 (vol 53 pp. 629)

Chinen, Naotsugu; Hosaka, Tetsuya
Asymptotic Dimension of Proper CAT(0) Spaces that are Homeomorphic to the Plane
In this paper, we investigate a proper CAT(0) space $(X,d)$ that is homeomorphic to $\mathbb R^2$ and we show that the asymptotic dimension $\operatorname{asdim} (X,d)$ is equal to $2$.

Keywords:asymptotic dimension, CAT(0) space, plane
Categories:20F69, 54F45, 20F65

5. CMB 2003 (vol 46 pp. 268)

Puls, Michael J.
Group Cohomology and $L^p$-Cohomology of Finitely Generated Groups
Let $G$ be a finitely generated, infinite group, let $p>1$, and let $L^p(G)$ denote the Banach space $\{ \sum_{x\in G} a_xx \mid \sum_{x\in G} |a_x |^p < \infty \}$. In this paper we will study the first cohomology group of $G$ with coefficients in $L^p(G)$, and the first reduced $L^p$-cohomology space of $G$. Most of our results will be for a class of groups that contains all finitely generated, infinite nilpotent groups.

Keywords:group cohomology, $L^p$-cohomology, central element of infinite order, harmonic function, continuous linear functional
Categories:43A15, 20F65, 20F18

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