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        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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        0answers
        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
        1answer
        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
        votes
        0answers
        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
        votes
        0answers
        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
        votes
        1answer
        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
        votes
        0answers
        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
        votes
        0answers
        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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        0answers
        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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        0answers
        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" ...
        3
        votes
        1answer
        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
        votes
        0answers
        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
        4answers
        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
        1answer
        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
        votes
        2answers
        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 ...
        9
        votes
        0answers
        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 $...

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