Papers

We study the word problem in two-generator one-relator inverse monoids, proving both decidability and undecidability results. We give the first example of a two-generator one-relator inverse monoid of the form $\mathrm{Inv} \langle a,b \mid w=1 \rangle$ with an undecidable word problem. Furthermore our examples can be chosen to be $E$-unitary. To prove this result we exploit the connection between the word problem for one-relator inverse monoids and the submonoid membership problem for one-relator groups established in work of Ivanov, Margolis, and Meakin (2001). We also give the first example of a two-generator one-relator group, with a reduced word defining relator, with an undecidable prefix membership problem. We investigate the problem of classifying the two-generator one-relator inverse mononids with decidable word problem. For this we prove a general result classifying when a free-by-cyclic group with a polynomially growing monodromy has decidable submonoid membership problem. Specifically we prove that a free-by-cyclic group with a polynomially growing monodromy has decidable submonoid membership problem if and only if it has decidable rational subset membership problem if and only if the defining automorphism has finite order in the outer automorphism group. We then apply that result to give a sufficient condition for a two-generator one-relator inverse monoid to have decidable word problem in the case its maximal group image is free-by-cyclic of polynomial growth, and we show that that condition can be algorithmically checked. Conversely, we show that when this condition is not satisfied then the two-generator one-relator inverse monoid can be lifted to an example with undecidable word problem and with the same maximal group image (up to taking a free product with the infinite cyclic group).

In this article, we investigate two notions of genericity for triples of antipodal ideal chambers in a locally finite affine building $X$: one defined at the ideal boundary $X^{\infty}$, which we call ideal-genericity, and the other defined from within the affine building \(X\), which we call affine-genericity. While ideal-genericity implies affine-genericity, the latter is the more suitable notion for constructing a barycenter map associated with affine-generic triples of antipodal ideal chambers. This perspective also allows us to establish that this barycenter map is locally constant, and hence continuous. Finally, we provide sufficient geometric conditions on the affine Weyl group associated with $X$ that guarantee both ideal- and affine-genericity for all triples of antipodal ideal chambers of $X^\infty$. These conditions yield an algorithmic method for constructing geometric configurations in $X \cup X^{\infty}$ (when they exist) that correspond to non-generic triples of antipodal ideal chambers of $X^{\infty}$. Furthermore, our computations for the irreducible finite Weyl groups of rank at least two show that automatic ideal-genericity holds for all triples of antipodal ideal chambers only in types $B_2 = C_2$, $G_2$ and $B_3$.

For a group $G$, let $S(G)$ be the set of elements $g \in G$ such that $\langle g,x\rangle$ is solvable for all $x \in G$. We study when $S(G)$ coincides with the locally solvable radical $R_{\mathrm{L}\mathfrak{S}}(G)$. Using Wilson's profinitely convergent word sequences, we show that, for every locally (solvable-by-finite) group $G$ and every Wilson sequence $\omega$, $$ R_{\mathrm{L}\mathfrak{S}}(G) = R_{\mathrm{L}\mathfrak{R}}(G) = S(G) = W_{\omega}(G).$$ We also obtain four-conjugate and seven-commutator descriptions of radical membership, together with a two-conjugate result for torsion elements of order coprime to $6$. The same radical identity holds for locally linear groups, and hence for subgroups of $\mathrm{GL}_{\infty}(D)$ when $D$ is a locally finite-dimensional division ring. Independently, we prove that groups with nearly modular subgroup lattice satisfy $$R_{\mathrm{L}\mathfrak{S}}(G) = R_{\mathrm{L}\mathfrak{R}}(G) = S(G).$$

We study the simplicial boundary $\partial\mathcal{FS}$ of Culler-Vogtmann Outer space via thickenings defined by graph-theoretic connectivity. Let $C'$ be the subcomplex of the free splitting complex obtained from $\partial\mathcal{FS}$ by adding all stable graphs that are not $3$-edge connected, together with their faces. We prove that the inclusion $\partial\mathcal{FS}\hookrightarrow C'$ is $(2n-3)$-connected. The proof shows, more precisely, that adding graphs with cut vertices is a homotopy equivalence, while the only non-contractible fibres in the $2$-bond thickening occur over $\theta$-graphs. The result gives further evidence that $\partial\mathcal{FS}$ may be $(2n-3)$-spherical, an $\operatorname{Out}(F_n)$-analogue of Rognes's connectivity conjecture for the common basis complex. It also gives a topological, universal-cover perspective that unifies several existing results about the commutative graph complex.

We study the functorial and categorical structure of the canonical rigid direct-system representation of finite local-unit-aligned totally ordered monoids. The local-unit map $\tau$ induces a canonical $\tau$-multiplication-coherent decomposition into component monoids, and the associated representation reconstructs the original ordered monoid from a finite chain-indexed rigid direct system whose proper transition maps are unit-constant. The present paper identifies the morphism classes for which this representation is categorical. First, we prove an equivalence between finite local-unit-aligned totally ordered monoids with strict block morphisms and rigid direct systems with directed-order-compatible system morphisms. Second, we prove an intrinsic equivalence for $\tau$-compatible homomorphisms, that is, isotone unital homomorphisms commuting with the local-unit map. In this second setting, distinct positive idempotents and hence distinct canonical components may collapse to a single target component; on the direct-system side this is represented by non-injective isotone index maps together with component maps satisfying the corresponding collapse and absorption axioms. Thus the canonical rigid direct-system representation is functorial both for strict block morphisms and for intrinsic $\tau$-compatible homomorphisms.

We generalize known results on normalized symmetric coordinates and their dual differential operators, established for the symmetric group, to the complex monomial reflection group $G(r,n):=\mu_r\wr S_n$ (with $G(1,n)=S_n$). The central tool, proved in detail, is a transfer principle: the substitution $y_i=x_i^r$ identifies the invariant ring of $G(r,n)$ with that of $S_n$ and transports, term by term, the corresponding operators and coordinates into explicit rational objects in the original variables $x_i$. From this we deduce the $G(r,n)$-analogues of the known results for $S_n$ existence and uniqueness of the dual coordinates $U_k$, and a Weyl algebra structure localized at the discriminant of $G(r,n)$ with complete and self-contained proofs for the points that do not follow directly from the transfer (Leibniz rule, main theorem). We then treat the total diagonal: its preimage splits into $r^{n-1}$ lines permuted transitively by the group, and there, unlike the transferred operators $\Delta_i$, the raw derivatives $\partial_{x_i}$ exhibit a phenomenon specific to $r\ge2$ that we describe completely via a Fa di Bruno-type structure formula. Finally, we give a closed formula for the constants of this structure formula (via Bell polynomials), completely resolve the degeneracy at an isolated point $x_i=0$ (the operator $\Delta_i$ extends holomorphically there, with $\Delta_i\phi=\partial_i^r\phi/r!$), and deduce from this a partial analogue of the description of the tangent space to the GIT quotient $\CC^n/G(r,n)$; the case of several coordinates vanishing simultaneously remains open and is precisely delineated. Full proofs of all new results are given in detail.

A group $G$ is invariably generated by two elements $a$ and $b$ if $G=\langle a^g,b^h\rangle $ for all $g,h\in G$. This paper provides a structural description of finite groups invariably generated by two elements of distinct prime orders $s$ and $t$. We illustrate the use of our main result in a case study with $s=2$ and $t=3$. In particular we prove that an almost simple group is invariably generated by an element of order $2$ and an element of order $3$ if and only if it is isomorphic to $\mathrm{PGL}_2(3^{2^b})$ for some $b \ge 1$.

A remarkable result of Guba shows that the Dehn function of Thompson's group $F$ is quadratic (Inv. Math. 163:313-342). In another paper (Alg. prob. in Grps. and Semigrps, 91-120), Guba showed that Thompson's group $T$ and $V$ have Dehn functions bounded above by $n^7$ and $n^{11}$, respectively. Recently, Migliorini showed that the Dehn function of Thompson's group $T$ is also quadratic (For. of Math. Sigma, 13:e109). In this paper, we improve the upper bound of the Dehn function of Thompson's group $V$ to $n^6$ and explore the intricacies of its relations via explicit van Kampen diagrams.

A group $\Gamma$ generated by a finite set $S$ has Property $(T)$ if the associated Kazhdan constant $\kappa(\Gamma,S)$ is positive. If so, the same holds for every finite generating set $S$. There are Property $(T)$ groups which are uniformly $(T)$, and others which are not. It is a well-known problem whether the classical examples of Property $(T)$ groups, $\mathrm{SL}_n(\mathbb{Z})$, $n\geq 3$, are uniformly $(T)$ or not. We show that they are not. Moreover, the same holds for every infinite finitely generated linear group.

We establish a simple characterization of amenability of second countable locally compact quantum groups $\mathbb{G}$ in terms of the long-time behavior of convolution semigroups of states on $\mathbb{G}$. We then investigate the deeper problem of combining amenability of $\mathbb{G}$ with other approximation properties of the dual quantum group $\hat{\mathbb{G}}$. Specifically, we characterize the combination of ("strong") amenability of $\mathbb{G}$ with either the failure of property (T) or the presence of the Haagerup property for $\hat{\mathbb{G}}$. These results, which appear to be new even in the classical setting of locally compact groups, produce convolution semigroups on $\mathbb{G}$ whose behavior is controlled both as time goes to zero and as time goes to infinity.

Casolo asked whether a torsion-free group in which every subgroup is subnormal of defect at most $n$ must be nilpotent of class at most $n$. The answer is known to be positive for $n\leq 4$. We construct a $2$-generated torsion-free nilpotent group $G$ such that $$ \cl(G)=6 \qquad\text{and}\qquad [G,{}_5H]\leq H \quad\text{for every }H\leq G. $$ Thus Casolo's question has a negative answer for $n=5$.

Let $r>1$ with $\gcd(r,30)=1$. We construct a finite group $G_r=(C_5\times C_3\times C_r)\rtimes W$, where $|W|=32$ and $G_r'\cong C_{60r}$, together with an augmentation-preserving automorphism $\alpha_r\in\operatorname{Aut}(\mathbb{Z}G_r)$ having no Zassenhaus factorization. The image $Y_r=\alpha_r(G_r)$ is a normalized group basis which is not rationally conjugate to $G_r$, although every element of $Y_r$ is individually rationally conjugate to an element of $G_r$. Consequently, (ZC2) and (ZC3) fail for finite cyclic-by-abelian groups, resolving a problem of Margolis and del Río. The construction extends Hertweck's example uniformly: both the class-preserving obstruction and the integral gluing are independent of the order of the auxiliary factor. The smallest admissible member of this family has order $3360$ and derived subgroup $C_{420}$.

We give a negative answer to Kourovka Notebook Problem 21.88: no finite group of odd order has commuting probability $1/17$. This follows from a structural theorem asserting that, whenever $p$ is an odd prime and $cp(G)=1/p$, a Sylow $p$-subgroup of $G$ is normal and abelian. Together with Burnside's congruence for the number of conjugacy classes of a group of odd order, this also excludes $cp(G)=1/p$ for every odd prime $p<97$. We further study the next unresolved case not excluded by this congruence, namely $p=97$.

Higman's conjecture that Z[G] has only trivial units, for G torsion-free, is open for the Promislow (Hantzsche-Wendt) group P, the group over which Gardam disproved the field-coefficient unit conjecture in 2021. We introduce an exact reduction of the integral conjecture for P modulo 2 into two sub-problems, record the base cases as consequences of the Craven-Pappas small-length theorems, and argue that the frontier of the integral problem is word-radius 4: a triviality theorem for Z[P] there would be the first statement separating Z from every field. Throughout, claims are ball-limited and stated as such; we make no claim on the full conjecture.

We prove that every finite group that is the product of every pair of its non-conjugate maximal subgroups is soluble, answering Problem 10.34 of the Kourovka Notebook. The almost-simple case was proved by Tikhonenko and Tyutyanov. We treat the remaining minimal-counterexample branch, where the unique minimal normal subgroup has the form S^k, with S nonabelian simple and k >= 2. Two automorphism-stable coordinate subgroup classes satisfying a maximal-supplement criterion produce non-conjugate maximal supplements. If the factorization hypothesis held, their product would impose a p-adic divisibility requirement growing linearly with k, while the quotient contributes only coordinate outer automorphisms and a factor dividing k!. A fixed valuation gap therefore excludes every k >= 2 at once. Suitable subgroup classes are constructed uniformly across the infinite families of finite simple groups using parabolic, torus, and primitive-prime-divisor arguments; stable flag parabolics handle graph fusion, while GAP certificates cover designated finite and sporadic cases. The resulting all-k obstruction provides a reusable mechanism for eliminating direct-power socles in finite-group factorization problems.

We show that, up to isometry, there is exactly one compact hyperbolic Coxeter 6-polytope with 10 facets, the polytope $P_{6,10}$ attributed to Bugaenko. Together with results of Felikson-Tumarkin ($d \ge 7$) and of Burcroff and Ma-Zheng ($d = 4, 5$), this completes the classification of compact hyperbolic Coxeter $d$-polytopes with $d+4$ facets. The proof is computer assisted. Affine Gale duality applied to the complete database of order types on 10 points yields 387 combinatorial types, of which Lannér's classification excludes 83. For the remaining 304, an exhaustive search over Coxeter labellings, with no a priori bound on the dihedral angles, leaves a single realizable Gram matrix. Every rejection is certified in exact arithmetic, and completeness of the search is certified independently by DRAT proofs checked by drat-trim. The same code reproduces the known censuses in dimensions 4 and 5. Code, data and certificates are publicly available.

We study sets of (measurable) returns in countable groups $G$, namely sets of the form $\{g\in G:\mu(A\cap T_gA)>0\}$ arising from measure-preserving actions. Extending a result of Bergelson, we show that sets of returns in $G\times G$ contain subsets of the form $B\times B$, where $B$ is large with respect to suitable notions of largeness that remain meaningful even for non-amenable groups. As a consequence, if $G$ is amenable, then every sufficiently large subset $A\subseteq G\times G$ satisfies $B\times B\subseteq AA^{-1}$ for some large set $B\subseteq G$. We also investigate when sets of returns in $G$ contain product sets $BB$ with $B$ large. In contrast with the Cartesian-product phenomenon above, this problem is considerably subtler in non-abelian groups and is closely connected to `symmetric correlation functions', namely functions of the form $g\mapsto \mu(T_g^{-1}A\cap T_gA)$. We use this connection to show that, for broad classes of amenable groups - including finitely generated nilpotent groups and certain solvable non-nilpotent groups, every sufficiently large set $A\subseteq G$ contains a large subset $B$ satisfying $BB\subseteq AA^{-1}$. Finally, we establish polynomial analogues of these results for finitely generated nilpotent groups, extending earlier work of Bergelson and Ruzsa.

We construct two explicit countable discrete groups $\Gamma_1$ and $\Gamma_2$ that are both ICC and have Kazhdan's property (T). Although $\Gamma_1$ and $\Gamma_2$ are not isomorphic as groups, their group von Neumann algebras are isomorphic: $L(\Gamma_1)\cong L(\Gamma_2)$. This provides a counterexample to Connes' rigidity conjecture for ICC property (T) groups. This result was obtained with assistance from GPT-5.6 Sol, independently of and concurrently with work by OpenAI.

This paper is part of a series devoted to the classification of isomorphisms and automorphisms of twisted Chevalley groups over commutative rings. In this work, we prove that every automorphism of a twisted Chevalley group of type ${}^2 D_\ell$ ($\ell \geqslant 4$) over a local ring containing $1/2$ is standard.

In [Publ. Mat. 67 (2023), no. 1, 173-198], G. Navarro asked whether the character field of an odd-degree character of a finite group is generated by a single character value. We show by explicit examples that the answer is negative in general.

A group word $w$ is said to be strongly concise in a class $\mathcal{C}$ of profinite groups if, for every group $G \in \mathcal{C}$ such that $w$ takes less than $2^{\aleph_0}$ values in $G$, the verbal subgroup $w(G)$ is finite. Using the notion of polynomial mappings introduced by Passi, we establish that every group word is strongly concise in the class of profinite metabelian groups. With this new approach, we also give an alternative proof for the fact that every group word is strongly concise in the class of profinite nilpotent groups.

We study the expected number of fixed points of a random permutation obtained via a word map, with surface group constraints imposed. For stable irreducible characters $\chi$ of the symmetric group $S_{n}$ and with $R_{g}=[a_{1},b_{1}]\dots[a_{g},b_{g}]$ and $w\in F_{2g}$, we compute $\mathbb{E}_{S_{n}^{2g}}\left[\chi\left(R_{g}(h)\right)\#\mathrm{fix}\left(w(h)\right)\right]$. We show that, if $w$ is a shortest representative for the conjugacy class of $\gamma\in\Gamma_{g}=\left\langle a_{1},b_{1},\dots,a_{g},b_{g}:R_{g}\right\rangle$, then this expectation is $O\left(1/\dim\chi\right)$. As an application, we recover a boundedness statement of Magee–Puder on the large $n$ limit of the expected number of fixed points of $\phi_{n}(\gamma)$, where $\gamma\in\Gamma_{g}$ is fixed and $\phi_{n}\in\hom\left(\Gamma_{g},S_{n}\right)$ is chosen uniformly at random.

This paper completely classifies when the pro-$p$ completion of an amalgamated product has bounded generation, assuming its starting vertex groups are already boundedly generated. We first prove that the pro-$p$ completion of any abstract amalgam always naturally forms a proper pro-$p$ amalgam. Within this framework, we show that bounded generation is extremely rare. Specifically, the completed group is boundedly generated if and only if one of the vertex groups completely collapses into the edge group (meaning it has an index of 1), or in the highly specific case where $p=2$ and both vertex groups have an index of exactly 2 over the edge intersection.Additionally, we develop a new set of criteria to determine when the original vertex and edge groups map into the completion without collapsing. Finally, we provide a cautionary counterexample: for any given prime $p$, we construct an amalgam whose pro-$p$ completion is boundedly generated, but its full profinite completion is not. This demonstrates that possessing bounded generation at a single prime does not guarantee that the full profinite group will share this property.

We prove a graph-theoretical characterisation of finitely generated subgroups of Thompson's group $V$: a finitely generated group embeds in $V$ if and only if it admits a faithful context-free action, or equivalently if it belongs to the class CF-TR of transition groups of context-free graphs recently introduced by Matucci and the three last authors. Using this characterisation, we prove results in different directions: - All known examples of groups with co-context-free Word Problem do embed in $V$, providing evidence towards Lehnert's conjecture. - Each finitely generated subgroup of $V$ is either virtually abelian, or contains a free non-abelian semigroup. It follows that groups of intermediate growth do not embed in Thompson's $V$. We further study the relation between transition groups defined by graphs that are limits or covers of each others, and prove properties of transition groups of context-free graphs of polynomial growth. Finally, we prove that the Basilica and Hanoï Towers groups do not embed in $V$. This uses the geometry of Schreier graphs of the natural actions of these groups and of Thompson's $V$.

OpenAI announced the existence of a non-sofic group: the unit group of the binary Leavitt algebra. We exhibit a different source of examples (relying on the same technical criterion) which includes torsion-free groups.

We prove that the Diophantine problem, i.e. the problem of solving equations with coefficients, in one-relator groups and one-relation monoids is undecidable in general. Specifically, we show that for all $d \geq 2$, the Diophantine problem for the Baumslag-Gersten group $\mathcal{B}_d = \operatorname{BG}(1,d)$ and for a closely related one-relation monoid $\mathcal{M}_d$ are both undecidable. Both results are obtained by an encoding of a recent undecidability result due to Rybalov.

For every odd prime $p$, we construct a finite left brace $A_p$ of order $p^{2p+1}$ whose additive group is elementary abelian and whose multiplicative group $G_p$ is a powerful $p$-group, but such that $A_p$ is not right nilpotent. More precisely, $G_p'=G_p^p\cong C_p^2, \operatorname{cl}(G_p)=2, \exp(G_p)=p^2,$ and $\operatorname{Soc}(A_p)=0$. Thus powerfulness does not force right nilpotence even for multiplicative groups of class two with derived subgroup of order $p^2$. The obstruction is explicit: $A_p$ contains a three-dimensional trivial ideal $T$ satisfying $T*A_p=T$, whereas its left series is the ideal-power filtration of a nilpotent commutative algebra. Both $T$ and $A_p/T$ are right nilpotent, so right nilpotence of finite left braces is not closed under extensions. The construction is uniform and arises from a finite local commutative algebra, a square-zero derivation, and an invariant group homomorphism, yielding a regular affine subgroup as the kernel of a homomorphism on a semidirect product. It disproves the Shalev–Smoktunowicz conjecture in every odd characteristic and yields finite non-degenerate irretractable involutive set-theoretic solutions of the Yang–Baxter equation whose permutation groups are powerful $p$-groups of class two.

For a finite group $G$, let $\omega(G)$ be the set of element orders of $G$ and let $h(G)$ be the number of pairwise nonisomorphic finite groups $H$ with $\omega(H)=\omega(G)$. We say that the recognition problem is solved for $G$ if the number $h(G)$ is known, and if it is finite, then all finite groups $H$ with $\omega(H)=\omega(G)$ are listed. We complete the solution of the recognition problem for all finite simple groups.

Let $\mathcal{H}$ be the Hurwitz quaternions, $p$ an odd prime, and $Q \in \mathcal{H}$ a prime of norm $q \neq p$. Metacommutation $PQ = Q'P'$ induces a permutation $\pi_Q$ of the $p+1$ left-associate classes of primes of norm $p$. Cohn and Kumar compute its sign and fixed-point count, and Leite and Machiavelo its full cycle structure, by formulas depending only on conjugation-invariant data of $Q$ (namely $q$ and $\mathrm{tr}\,Q$). We prove this is exactly the boundary of what such invariants can carry: no quantity $I(Q)$ invariant under unit conjugation determines $\pi_Q$ as a labelled permutation of the intrinsic class set, or even the image of a single specified class. The proof combines an equivariance identity $\pi_{uQu^{-1}} = \rho_u \pi_Q \rho_u^{-1}$ with a minimal, fully explicit witness at $(p,q) = (3,5)$: the four primes $2+i$, $2+j$, $2+k$, $2-i$ form a single unit-conjugacy orbit, hence agree under every conjugation-invariant function, yet induce four pairwise distinct $4$-cycles of the same four classes. We further observe that isomorphisms $\mathcal{H}/p\mathcal{H} \to M_2(\mathbb{F}_p)$ form a torsor under $\mathrm{PGL}_2(\mathbb{F}_p)$, so no projective labelling of the classes is canonical, and that by orbit-stabilizer the datum of one destination $\pi_Q(C)$ is exactly a coset $g_Q G_C$ in $\mathrm{PGL}_2(\mathbb{F}_p)/G_C$. All sixteen refactorizations in the witness are listed in the appendix and have been verified by machine along two independent routes.

Let $k$ be an algebraically closed field, and let $G$ be an algebraic $k$-group. We study finite abelian $k$-subgroups $A \subset G$ whose order is not divisible by the characteristic of $k$. This is a classical topic in the theory of algebraic groups going back to the work of Borel in the early 1960s. We sharpen previously known results on the structure of $A$. In particular, we show that there exists a maximal torus $T$ of $G$ such that the index $[A: (A \cap T)]$ divides the Grothendieck torsion index $t(G)$. We also show that there exists a maximal torus $T$ such that the quotient group $A/(A \cap T)$ is “small” in a suitable sense. As applications of these results, we (i) give a positive answer to a question of Totaro for $G$-torsors over fields $k_r = k((t_1))((t_2)) \ldots ((t_r))$ of iterated Laurent series, (ii) prove a variant of the “hypothèse optimiste” of Tits about splitting fields of $E_8$-torsors, and (iii) show that certain torsors over $k_r$ cannot be split by the function field of a genus $1$ curve.