Let $G$ be a torsion-free one-ended hyperbolic group and let $\phi\in\operatorname{Aut} (G)$. We prove that the mapping torus, or suspension, $$ M:=G_{\phi}=G\rtimes_\phi \mathbb{Z} $$ has solvable conjugacy problem. This builds on the pioneering work of Pr\'eaux who solved the conjugacy problem for all (geometrisable) three-manifolds. We view $M$ as a generalisation of a fibred three-manifold. Our proof begins with the canonical JSJ tree of $G$, whose suspension gives a graph-of-groups decomposition of $ M $. We refine each suspended QH vertex using a Nielsen--Thurston reduction system for its induced monodromy, taking account of non-orientable surfaces and orientation-reversing monodromy. Equivalently, this is a further geometric JSJ decomposition for each vertex which is a fibred three-manifold, but allowing Klein bottles as well as tori in the splitting. We then `block'pieces of this refined JSJ together according to whether they share a central element, glued along an elementary vertex, by a sequence of folding operations. This new splitting is cocompact and acylindrical; in particular certain local `solution sets'turn out to be rational subsets of virtually abelian subgroups - the virtually abelian groups in question are products of the edge groups. Our version of Pr\'eaux's graph-of-groups argument then reduces global conjugacy to effective intersection and non-emptiness for these rational sets.
Let $H_\Gamma$ be the Bestvina-Brady group associated to a finite connected graph $\Gamma$. For a biconnected defining graph, we prove two structure theorems. First, restriction induces an isomorphism $\mathrm{IAut}(A_\Gamma)\cong \mathrm{IAut}(H_\Gamma)$ compatible with the Andreadakis-Johnson filtrations. Second, the quadratic and cubic lower-central relation spaces, together with the separator arrangement detected by the Bieri-Neumann-Strebel invariant, determine a rational associative algebra $\mathscr{C}_\Gamma$. Every integral rank-one square-zero element of this algebra is realized by an automorphism of $H_\Gamma$, and the subgroup generated by these roots has finite index both in the cohomological image of $\mathrm{Aut}(H_\Gamma)$ and in the unit group of an integral order in $\mathscr{C}_\Gamma$. For an arbitrary connected graph, the graph-block decomposition gives the Grushko decomposition of $H_\Gamma$. Relative free-product automorphism theory then implies that $\mathrm{Aut}(H_\Gamma)$ and $\mathrm{Out}(H_\Gamma)$ are finitely generated and satisfy the Tits alternative relative to virtually polycyclic groups. We prove that $\mathrm{Aut}(H_\Gamma)$ is finitely presented if and only if $\mathrm{Out}(H_\Gamma)$ is finitely presented. This equivalence fails for higher finiteness properties without additional hypotheses: for $\Gamma_m=C_m\vee K_3$ with $m\geq 5$, $\mathrm{Out}(H_{\Gamma_m})$ is of type $F_\infty$, whereas $\mathrm{Aut}(H_{\Gamma_m})$ is of type $F_3$ but not $F_4$. We also construct a type-$F_\infty$ Bestvina-Brady group whose automorphism and outer automorphism groups are finitely generated but not finitely presented, and show that $H_{C_n}$ is not finitely presented for $n\geq 5$, whereas $\mathrm{Out}(H_{C_n})$ is virtually infinite cyclic.
In this work, we construct a generalization of the $G_n^k$-theory to the case of an arbitrary hypergraph. The case of spherical braids is considered separately, using the stratification of the moduli space $\mathcal{M}_n(S^2)$ and the hypergraph $\Gamma_n^{\mathrm{sph}}$ encoding projective constraints. In contrast to the original $G_{n}^{k}$ theory, where codimension-one properties are determined by exactly $k$ particles, the present work considers various cases corresponding to strata of codimension~$1$. These groups admit nice maps to free products of cyclic groups. Among unsolved problems, we emphasize the question how the above construction works for abelian varieties and, in particular, for elliptic curves.
We prove M\"uhlherr's Twist Conjecture: any two angle-compatible Coxeter generating sets of a Coxeter group differ by a finite sequence of elementary twists and a conjugation. Combined with earlier work of Howlett-M\"uhlherr and Marquis-M\"uhlherr, this completes the resolution of the Isomorphism Problem for Coxeter groups. A further consequence is that ${\rm Aut}(W)$ is finitely generated for every Coxeter group $W$, and there is an algorithm producing a finite set of generators for ${\rm Aut}(W)$ starting from any Coxeter matrix. Of the vast literature on the Twist Conjecture, we utilise only two results in an essential way: strong rigidity of $2$--spherical Coxeter systems, due to Caprace and M\"uhlherr, and the framework of markings and hierarchies developed by Caprace and Przytycki for the twist-rigid case. We also exploit in a fundamental way some soft ideas from JSJ theory and an observation of Mihalik-Tschantz on splittings of Coxeter groups. No form of AI was used in the writing of this manuscript, nor in the research that it presents.
We construct random open covers of higher-rank locally symmetric spaces using a construction we call scaffolded Poisson processes. Let $X=G/K$ be a symmetric space of noncompact type and real rank at least $2$. We prove a general vanishing theorem for the normalized torsion in first homology along sequences of torsion-free lattices in $G$. In particular, if $G$ is simple, we get \[ \dfrac{\log |H_1(M_n;\mathbb{Z})_{\operatorname{tors}}|}{\mathrm{vol}(M_n)} \longrightarrow 0 \] for any sequence of distinct manifolds $M_n = \Gamma_n \backslash X$. This answers a question of Ab\'ert, Gelander, and Nikolov, and confirms the degree-one vanishing with trivial integral coefficients predicted by a conjecture of Bergeron and Venkatesh in the higher-rank setting. In addition, we get quantitative bounds with respect to the minimal injectivity radius for both the torsion in first homology and the minimal number of generators of $\Gamma$. Finally, we prove the analogous statements for affine buildings.
The fundamental groups of compact 3-manifolds are known to be residually finite. Feng Luo conjectured that a stronger statement is true, by only allowing finite groups of the form $\mathrm{PGL}(2,R)$, where $R$ is a finite commutative ring. In earlier work, this conjecture was disproven in full generality. The conjecture arose in the context of orientable connected compact 3-manifolds which are geometrizable. By constructing explicit faithful linear representations using rings with nilpotent elements, we demonstrate that the conjecture holds for six of the eight Thurston model geometries, namely all but $\mathbb{S}^3$ and $\widetilde{\mathrm{SL}_2}$. In the case of $\mathbb{S}^3$, the conjecture holds if we replace $\mathrm{PGL}(2,R)$ with $\mathrm{GL}(2,R)$. A spherical counterexample for the projective variant is the Poincar\'{e} homology sphere $\Sigma(2,3,5)$. In the case of $\widetilde{\mathrm{SL}_2}$, the conjecture fails to hold for both the projective and non-projective variants; a counterxample is provided by the Brieskorn sphere $\Sigma(2,3,7)$.
Let $G$ be an affine algebraic reductive group over $\mathbb{C}$ whose identity component is a torus $T$, and let $K:=G/T$. We realize $\Rep(G)$ as an equivariantization $\Rep(T)^K$ and describe its finite indecomposable semisimple module categories in terms of equivariant module-category data over $\Rep(T)$. For such an equivariantization, rank one is characterized by transitivity of the induced action on the simple objects of the underlying $\Rep(T)$-module category, and nondegeneracy of a projective cocycle on a point stabilizer. We use this criterion to parametrize fiber functors on $\Rep(G)$, prove that every such fiber functor is classical, hence arises from a Hopf $2$-cocycle on $\mathscr{O}(G)$, and classify (minimal) Hopf $2$-cocycle on $\mathscr{O}(G)$. For commutative direct products $G=T\times K$, we also give the canonical K\"unneth decomposition of the group of gauge classes, including the mixed component, and compare it with our classification.
Shlomo Gelaki· 0 citations
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