Large scale properties of groups; growth functions; Dehn functions; small cancellation properties; hyperbolicity and CAT(0); actions and representations; combinatorial group theory; presentations

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79 views

### Can every non-inner automorphism of a group with residually finite outer automorphisms be realised as an non-inner automorphism of a finite quotient?

First some motivation: most proofs that show that the group of outer automorphisms is residually finite do not only show that the subgroup of inner automorphisms is closed in the profinite topology, ...

**4**

votes

**1**answer

118 views

### Infinite finitely presented simple group (or more generally with trivial profinite completion) that is not amalgamated free product

As is described in the title. Is there a finitely presented group $G$, with trivial profinite completion $\widehat{G}=0$, which is not amalgamated free product?
For example, the famous example Higman ...

**6**

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57 views

### Stable commutator lengths of pseudo-Anosovs

Does anyone have an example of a pseudo-Anosov mapping class for which the stable commutator length is known exactly?

**4**

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**0**answers

112 views

### Non-algebraic quasi-isometric embeddings

What are examples of finitely generated groups $\Gamma$ and $\Lambda$ such that the metric space $\Lambda$ embeds into $\Gamma$ quasi-isometrically but such that $\Lambda$ is very much not a subgroup ...

**10**

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**1**answer

280 views

### Residually finite group surjective to nonresidually finite group with finitely generated kernel

As is described in the title, is there a known example such that there is a surjective homomorphism of groups $$f: G\rightarrow H,$$ with $G$ and $H$ finitely presented, $G$ is residually finite, and $...

**5**

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**0**answers

95 views

### Automorphism groups of cocompact Fuchsian groups as mapping class groups

Let $\Gamma$ be a cocompact Fuchsian group. So it has presentation $$\langle x_1,y_1, \dots, x_g,y_g,z_1, \ldots, z_r \mid [x_1,y_1] \cdots [x_g,y_g]z_1 \cdots z_r=1, \ z_i^{m_i}=1 \rangle$$
for some $...

**4**

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136 views

### Uniqueness of the boundary of a hierarchically hyperbolic group

Hierarchically hyperbolic groups and spaces (HHG and HHS for short) were defined by Behrstock, Hagen and Sisto (see here and here). Examples include mapping class groups, Right angled Artin groups, ...

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54 views

### Which $CAT(0)$-polygonal complexes are median spaces?

$CAT(0)$-polygonal complexes are simply connected collections of glued (on their faces) polyhedra of varying dimension such that each link is flag.
Which $CAT(0)$-polygonal complexes with appropriate ...

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**0**answers

56 views

### Do $CAT(0)$-polygonal complexes have hyperplanes?

$CAT(0)$-polygonal complexes are simply connected collections of glued (on their faces) polyhedra of varying dimension such that each link is flag.
Do $CAT(0)$-polygonal complexes have "hyperplanes" ...

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**1**answer

123 views

### Are $CAT(0)$-polygonal complexes median spaces?

A median space is a metric space $X$ for which for any three points $x, y , z \in X $ there exists a unique point $m$ such that $d(x,m)+ d(m, y)= d(x , y ), d(x,m)+ d(m, z)= d(x , z ), d(y,m)+ d(m, z)=...

**5**

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123 views

### The preimage of bounded real intervals under homomorphisms on hyperbolic groups

Let $G$ be a hyperbolic group with a fixed (finite, symmetric) generating set and suppose that $\varphi : G \to \mathbb{R}$ is a group homomorphism. Write $W_n = \{ g \in G: |g|=n\}$, where $|g|$ ...

**6**

votes

**4**answers

500 views

### What is a geodesic in Outer space?

The Culler-Vogtmann Outer space $\text{CV}_n$ is an analogue of Teichmuller space for the group $\text{Out}(F_n)$.
Is there any notion of a geodesic path in $\text{CV}_n$? Are there different ...

**2**

votes

**1**answer

245 views

### About the growth rate of a group

Let $G$ be a f.g. group and $d$ be a word metric w.r.t. a symmetric generating set. For $g\in G$, define $|g|:=d(g,e)$, where $e$ is the group identity. For $k\in\mathbb N$, put
$$n_k:=\#\{g\in G: |g|...

**5**

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**2**answers

170 views

### Codimension-1 subgroups of 3-manifold groups

Let $G$ be a finitely generated group and let $H$ be a subgroup of $G$. $H$ is a codimension-1 subgroup of $G$ if $C_{G}/H$ has more than one end, where $C_{G}$ is the Cayley graph of $G$.
Do all ...

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**0**answers

237 views

### Sharpness of the $1/6$-constant in the Cancellation Theorem

I originally posted this question over at Stackexchange, before realising it is much better suited for Overflow:
Let $\langle \; S \; | \; R \; \rangle$ be a presentation of a group $G$ with a set $...