Questions tagged [geometric-group-theory]
Large scale properties of groups; growth functions; Dehn functions; small cancellation properties; hyperbolicity and CAT(0); actions and representations; combinatorial group theory; presentations
319
questions with no upvoted or accepted answers
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378
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Discontinuity of Radon-Nikodym derivative for Patterson-Sullivan measures for word metrics on Gromov hyperbolic groups
Let $\Gamma$ be a Gromov hyperbolic group coming endowed with a word metric coming from some finite generating set. Let $\nu$ be an associated Patterson-Sullivan measure (quasi-conformal density).
I ...
3
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103
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Geometric automorphism of free group respect to nonorientable suface
An outer automorphism $[ϕ]\in Out(F_n)$ is geometric if it is induced by a surface homeomorphism h:M→M, where M is a compact surface with nonempty boundary. I am wondering is it enough we only ...
3
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206
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Growth of the number of generators in hyperbolic groups
Let $G$ be an infinite hyperbolic group, and let us further assume that it is residually finite (or even LERF/GFERF) so that we have plenty of subgroups of finite index.
I would like to know if one ...
3
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154
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Cancellations in products of two elements of a hyperbolic group
Let $G$ be a non-abelian free group with the standard generating set and the corresponding word metric. If we take two elements $g,h\in G$ and compute their product $gh$, some letters might cancel, ...
3
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297
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Induced graphs of Cayley graph
I have a Cayley graph $\mathrm{Cay}(G,S)$, its group presentation $G=\langle S | R \rangle$, and it becomes a metric graph by assigning a length equal to $1$ to each edge. I also have an induced ...
2
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241
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Interpretation of Kazhdan T property cohomologically
$\newcommand{\triv}{\mathit{triv}}$A group $G$ has property $T$ if $\triv$ is isolated in the space of unitary representations with the Fell topology.
In general, we heuristically have $H^1(G,Ad(V))$ (...
2
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97
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Orthogonal representation of free products of two groups
Suppose $A$ and $B$ are two countable, discrete, amenable groups. One definition of amenability tells us that there is a sequence of finitely supported, positive definite functions that converges to 1 ...
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131
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Need help understanding the geometry of a particular building structure
$\DeclareMathOperator\SL{SL}$I’m not primarily a geometer, so apologies if this question is worded poorly. I’ve been looking at asymptotic cones of connected semisimple Lie groups with at least one ...
2
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57
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upper bound for the exponential conjugacy growth rate for non-virtually nilpotent polycyclic groups
Given $n ≥ 0$, the conjugacy growth function $c(n)$ of a finitely generated group $G$, with respect to some finite generating set $S$, counts the number of conjugacy classes intersecting the ball of ...
2
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148
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The growth rate of the group $\mathbb{Z}[1/2] \rtimes _\phi \mathbb{Z}$, where $\phi (1)$ corresponds to multiplying every number by $2$
Consider the group $G = \mathbb{Z}[1/2] \rtimes _\phi \mathbb{Z}$, where $\mathbb{Z}[1/2] = \{j/2^m \mid j \in \mathbb{Z}, m\in\mathbb{N} \}$, the dyadic rationals, and for every $n\in \mathbb{Z}$, $...
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133
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Strong converse of Kazhdan's property (T)
In his 1972 paper Sur la cohomologie des groupes topologiques II, Guichardet proved$^\ast$ that (non-abelian) free groups satisfy the following strong converse of property (T): The $1$-cohomology $H^1(...
2
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120
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Further questions to limit groups and an article of Fujiwara and Sela
I already have asked a question to the following article:
Koji Fujiwara, Zlil Sela, The rates of growth in a hyperbolic group, Invent. math. 233 (2023) pp 1427–1470, doi:10.1007/s00222-023-01200-w, ...
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137
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Proof of Zimmer's cocycle super-rigidity theorem
I was reading the proof of Zimmer's cocycle super-rigidity theorem from the book 'Ergodic theory and semi-simple groups' by Robert Zimmer (Theorem 5.2.5, page 98). But I am not able to understand it. ...
2
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162
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Commuting conjugate elements in torsion-free groups
I have come across the following question while studying projective modules over integral groups rings of torsion-free groups.
Given a non-unit $x\in G$ a torsion-free group, does there exist $g\in G$ ...
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80
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Property A, Higson-Roe condition and its applications
Recently I have been studying amenability of groups and property A, and I came across the Higson-Roe condition:
Let $X$ be a uniformly discrete metric space with bounded geometry. $X$ has property $A$ ...
2
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143
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Can distinct meridians commute in a knot group?
Suppose I have a knot $K$ in $S^3$. Given a diagram $D$ of $K$ I get the Wirtinger presentation $\langle x_1, \dots, x_a \mid r_1, \dots, r_c\rangle$ of its knot group $\pi(K) = \pi_1(S^3 \setminus K)$...
2
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117
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Convex subsets in abstract groups
Consider a group $G$. A norm on $G$ is a function $\|\cdot\|\colon G\to\mathbb R_+$ with (*) $\|gh\|\le\|g\|+\|h\|$ and $\|g\|=\|g^{-1}\|$ and $\|1\|=0$; the space of norms is a convex in $\mathbb R_+^...
2
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166
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Characterization of growth in terms of coarse algebraic topology
$$
\newcommand{\mc}[1]{\mathcal{#1}}
\newcommand{\mbb}[1]{\mathbb{#1}}
\newcommand{\opn}[1]{\operatorname{#1}}
\DeclareMathOperator\cap{cap}
\def\sse{\subseteq}
$$
Coarse spaces
Let $X$ be a coarse ...
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144
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"Hyperbolic rigidity of higher rank lattices" by Thomas Haettel
$\DeclareMathOperator\MCG{MCG}\DeclareMathOperator\CC{C}$In the article "Hyperbolic rigidity of higher rank lattices", Thomas Haettel has proved the following theorem: Let $\Gamma$ be a ...
2
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100
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Definition of the category QMet of metric spaces and quasi-isometries
I am following Clara Löh's Geometric Group Theory. An Introduction, and in remark 5.1.12, she defines the category QMet whose objects are metric spaces and whose morphisms are quasi-isometric ...
2
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106
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Moment of the hitting measure of a subgroup
Given a [finitely generated] group $G$ and a finite generating set $S$, a measure $\mu$ will have finite $\alpha$-moment if $\sum_{g \in G} \mu(g) |g|_S^\alpha$ is finite (where $|g|_S$ is the word ...
2
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67
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Uniqueness of JSJ deformation space when there are no QH vertices
Let $G$ be a finitely generated group and $\mathcal{A}$ a collection of subgroups closed under conjugation and taking subgroups.
I'm interested in the problem of classifying all deformation spaces of ...
2
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127
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Estimate word-metric length in free nilpotent groups
I would like to estimate the length of a word in a free nilpotent group. As the first example, I would like to estimate the word metric in the Heisenberg group $H_3$. This is the group of upper ...
2
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124
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Image of the pure braid group under the Artin presentation into the automorphism group of the nilpotent quotient of a free group?
As I know, it is unknown that the image of the mapping class group of the surface and its Johnson filtration under the higher Johnson homomorphisms.
There are a relationship between the mapping class ...
2
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62
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Quasi-isometry of solvable minimax groups
[Edits in brackets]
Consider two finitely generated solvable minimax groups $G_i$ ($i = 1,2$) so that $1 \to N_i \to G_i \to Z_i \to 1$ [splits]
with $N_i$ nilpotent, $Z_i$ infinite cyclic and $G_i$ ...
2
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78
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A quasi-isometric embedding of a convex cocompact subgroup
I am currently reading a paper where they state the following claim:
"For a convex cocompact representation $\rho: \Gamma \to G$, the existence of a cocompact invariant convex set $\mathcal{C}$ ...
2
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198
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Algebraic rigidity in the automorphism group of the Cantor set
Let C be a Cantor set (middle third). Now we know that C is a totally disconnected compact topological space with the natural topology (i.e., $C=\{0,1\}^{\mathbb{N}}$). Let G:=Homeo(C) be the set of ...
2
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193
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Is every cocompact lattice in Sp(n,1) residually finite?
We know that every finitely generated linear group over a commutative ring is a residually finite group by Mal'cev's theorem, but each cocompact lattice in Sp(n,1) is a finitely generated linear group ...
2
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137
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Term and theories about "relation-free" elements in a group?
For a group $G$, there are two elements a, b which are "relation-free",
i.e., there is no nonempty, reduced word $W(X,Y)$ such that $W(a,b)=1$ in $G$.
Is there any terminologies or theories ...
2
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187
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$V$-like actions of $V$
This continues my question about prefix-continuous bijections (since the answer was "yes").
Notation and conventions: Let $A$ be a finite alphabet and $L \subset A^*$ a language. Let $G$ be a group. ...
2
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126
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Finitely generated uniformly amenable groups
Keller in "Amenable groups and varieties of groups" introduces uniformly amenable groups as groups such that
there is a function $a: ]0,1[ \times \mathbb{N} \to \mathbb{N}$ such that
for any finite ...
2
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118
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Examples of groups admitting a proper $1$-cocyle for a bounded representation
A representation $\pi: G \to B(H)$ of a group $G$ on a Hilbert space $H$ is called bounded iff $\sup_{g \in G} \| \pi(g) \|_{B(H)} = C < \infty$. A $1$-cocycle with respect to the representation $\...
2
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229
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Finite index subgroup of HNN extension
Let $GX$ be a tree group (a right-angled Artin group such that the graph is a tree), such that not all the subgroups of $GX$ are necessarily RAAGs, so the length of the tree is greater or equal than $...
2
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88
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Is there a notion of "graph of bundles" analogous to a graph of groups?
Since the notion of a graph of groups relies mostly on the pushout, can we construct graphs of objects in some other category, say, vector bundles? If this is the case and we have a "fundamental ...
2
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162
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Can the 2-complex associated to a finitely presented group be triangulated?
Let G be a finitely presented group. K is the 2-complex associated to G which is constructed as taught in Algebraic Topology. That is , 1-cells corresponding to generators and 2-cells corresponding to ...
2
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86
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Hausdorff dimension of radial limit sets for divergence type subgroups
Let $X$ be a proper $CAT(-1)$ space.
Let $\Gamma<Isom(X)$ be a subgroup of divergence type.
Is it true that the Hausdorff dimension of the radial limit set of $\Gamma$ in $\partial X$ is equal to ...
2
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85
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Banach density of a sequence of spheres in a virtually nilpotent group
Let $G$ be a finitely-generated group of polynomial growth equipped with the word metric (with respect to a fixed symmetric generating set).
Let
\begin{equation*}
A = \left\{ g \in G: |g| = mn, n \...
2
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139
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Concentration of Reduced words
This might be a rather broad question, and I'll be satisfied with some intuition and pointers to relevant literature. However, I'll certainly fill in more context and details based on any feedback.
...
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85
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Automorphisms of a free topological product
Let $G$, $G_1$, $G_2$ be Hausdorff topological groups. I am mainly interested in the case when those groups are profinite.
Let $G$ act continuously on $G_1$ and $G_2$ via continuous automorphisms, i.e....
2
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93
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Change of generators and shortest product in groups
Let $G$ be a finitely generated group.
For a set of generators $B$ of $G$, $\ell_B(x)$ is the length of the smallest sequence of elements(and inverse of the elements) in $B$, such that the product ...
2
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131
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Group on 2 generators and greedy relations that preserve exponential growth
I'm not sure if there's a specific question here, other than perhaps "is this object studided" / "is there a better way of looking at it"; if it's too vague for this forum I apologize.
First take the ...
2
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138
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Equivariant meromorphic functions
Let $G \subset \rm{PSL}_{2}\mathbb{C}$ be a subgroup of the Mobius group of the 2-sphere $S^2$, and suppose that $G$ also acts on a second surface $M^2$ by automorphisms.
Does there exist a ...
2
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100
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Cellular basis of two isomorphic group algebras
Let $W_1, W_2$ be two groups and $\varphi: W_1 \to W_2$ a group isomorphism. For a commutative Noetherian integral domain $R$, we have a $R$-linear isomorphism $\overline{\varphi}: R[W_1] \to R[W_2]$.
...
2
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162
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Convexity of length function for surfaces with boundary
In the paper "The Nielsen realization problem" (here), Kerckhoff proved that the length function on the Teichmüller for closed surface is convex. In his paper "Geodesic length functions and the ...
2
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142
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example of fuchsian groups acting on 2-sphere by G. Martin
Currently I am reading a paper "Infinite group actions on spheres" by Gaven Martin. I am a first year graduate students and I got lots of questions, so one of them is about the following example: (...
2
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135
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Extending continuous functions from $\partial X$ to $X\cup \partial X$
Consider a proper geodesic hyperbolic space $X$ (in the sense of Gromov). Let $\partial X$ be its Gromov boundary. Consider a complex-valued continuous function on the boundary $f\colon\partial X\to\...
2
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208
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Extra large spherical joins
If $X$ and $Y$ are piecewise spherical complexes, then their spherical join $X * Y$ is CAT(1) if and only of $X$ and $Y$ are CAT(1) (see the appendix of the first Charney-Davis paper below). One of ...
2
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90
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Fully residually free groups and completion
Let $G$ be a fully residually free group with a finitely generated profinite completion. Is $G$ necessarily finitely generated?
2
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142
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Rank gradient in free products amalgamating a finite subgroup
Let $A,B$ be finitely generated groups with a common finite subgroup $C$. Suppose that $[A : C] > 2, [B : C] > 1$.
Must $A *_C B$ have positive rank gradient?
See Which 3-manifolds have ...
2
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123
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Salvaging Howson's theorem for free profinite groups
This is an attempt to find a correct version of Do free profinite groups satisfy Howson's theorem?
Let $F$ be a free profinite group, and let $A,B \leq F$ be closed finitely generated subgroups ...