Papers

We study the equational theory and subvariety structure of the ai-semiring variety $\V_\infty$ generated by all flat semirings $S(a_1\cdots a_k)$, where the letters \(a_i\) are pairwise distinct. Using interval semirings of posets, we characterize its subdirectly irreducible members and describe variety membership in terms of jointly separating families of strict order-preserving maps. We obtain explicit finite identity bases for \(\V_\infty\) and each $\V_k$ generated by $S(a_1\cdots a_k)$. Consequently, every flat semiring \(S(W)\) associated with a nonempty set \(W\) of linear words is finitely based. This yields finitely based ai-semirings with exactly \(k\)-nilpotent multiplicative reduct for each \(k\geq 1\). For each \(k\geq 1\), let \(\B_k\) be the subvariety of \(\V_\infty\) defined by the \((k+1)\)-nilpotent identity. We prove that each \(\B_k\) is generated by a finite interval semiring and that every proper subvariety of \(\V_\infty\) is contained in some \(\B_k\). The variety \(\B_3\) is a Cross variety with exactly \(11\) subvarieties, whereas \([\V_k,\B_k]\), \([\V_k,\V_{k+1}]\), and \([\B_{k-1},\B_k]\) each contain continuum many subvarieties for every $k\geq 4$. In particular, this provides infinitely many finitely based finite semirings $S(a_1\cdots a_k)$ whose generated variety has continuum many subvarieties for every \(k\geq 5\).

We give a criterion ensuring that sufficiently high sublevel sets of a discrete Morse function on a $\mathrm{CAT}(0)$ cube complex have polynomial integral homological filling functions. As one application, we verify this criterion for the Stein–Farley complexes of the Higman–Thompson groups $F_{d,r}$, $T_{d,r}$, and $V_{d,r}$. Consequently, these groups have polynomial homological filling functions in every degree $k\geq2$.

The lampshuffler group of $\mathbb{Z}$ is the semidirect product $\operatorname{FSym}(\mathbb{Z}) \rtimes \mathbb{Z}$, which consists of all permutations of $\mathbb{Z}$ that act as a translation outside a finite set. This infinite permutation group naturally contains as subgroups the wreath products $H \wr \mathbb{Z}$ for every finite group $H$. We prove that the Subgroup Membership Problem, and more generally, the Submonoid Membership Problem, are decidable in $\operatorname{FSym}(\mathbb{Z}) \rtimes \mathbb{Z}$. Our proof reduces Subgroup and Submonoid Membership in $\operatorname{FSym}(\mathbb{Z}) \rtimes \mathbb{Z}$ to Subgroup Membership in the wreath products $H \wr \mathbb{Z}$, which was shown to be decidable by Lohrey, Steinberg and Zetzsche (2015).

We construct explicit off-diagonal Ramsey graphs from polynomial families of incident point–hyperplane flags. Universal interpolation translates the exact dimension of ordered clique configurations into a bound on their coefficient–label incidence. At the critical dimension, a polynomial separates the forbidden-pair image from its diagonal; deleting the corresponding edges preserves a triangular rank certificate. Restriction of scalars realizes the rational endpoint without rounding loss. The resulting fixed-$s$ exponent has leading scale $s/(2\log_2s)$, and short Frobenius relations give the explicit example $R(16,t)\ge\Omega(t^{2.0539221767\ldots})$. A finite-fiber construction gives the range $s^2(\log s)^2=o(\log t)$. For fixed parameters, algebraic preprocessing terminates and field initialization, vertex decoding and adjacency take deterministic time polynomial in the extension degree. We also retain an elimination-free variant and formulate the filtering argument for general forbidden configurations with a distinguished edge.

Subset currents on free groups, on surface groups and, more generally, on word-hyperbolic groups, were introduced by Kapovich and Nagnibeda, and further studied by Sasaki. Subset currents extend ordinary geodesic currents and provide measure-theoretic generalizations of conjugacy classes of infinite quasiconvex subgroups. Their supports can be viewed as “branching laminations". For free groups and surface groups, we construct non-rational subset currents with genuinely branching supports that carry no other subset currents except scalar multiples. For every nonabelian finite rank free group and every closed hyperbolic surface group, we show that such a current can be chosen so that its support is uncountable, is disjoint from the locus of two-point boundary subsets, and consists entirely of Cantor subsets of the boundary. More generally, we prove that every nonempty metrizable Choquet simplex occurs as a compact convex base for the cone of currents carried by a minimal genuinely branching subset lamination. The same universality phenomenon extends, apart from the usual rational-current terminology, to every non-elementary torsion-free hyperbolic group. We also study finite-pattern complexity. We prove that if $R_L(n)$ counts the allowed round patterns of radius $n$ in a minimal subset lamination $L$, then $R_L(n)=O(n)$ forces the carried-current cone to be finite-dimensional, while infinite-dimensionality forces $R_L(n)/n\to\infty$; this universal threshold is sharp. We show that the coarse growth type of $R_L$ is independent of the chosen free basis and, more generally, of the chosen finite marked graph. We also show that a refined switch system recovers the exact dimension of the carried-current cone.

A group is an {\it $\SJ$-group} if it admits a general abelian series all of whose terms are subnormal of finite defect in the whole group. We construct a group $G$ which is not an~\hbox{$\SJ$-group}, although every countable subgroup of $G$ is an $\SJ$-group.

In this paper, we compute the orders of upper triangular symplectic Renner monoids using admissible sets. We obtain both recursive and closed-form formulas for these orders. We then derive their exponential generating functions and establish connections between these functions and the Laguerre polynomials.

We show how to solve the conjugacy (search) problem and the problem of determining whether a given element is periodic in symmetric diagram groups based on finite semigroup presentations.

We obtain presentations of Borel subgroups of certain Chevalley groups and Kac-Moody groups of simply laced type. More precisely, we show that Borel subgroups of simply laced Chevalley groups over a commutative, unital ring (satisfying a minor technical condition), are the amalgamated product of their rank 2 subgroups, generalising a result of Tits over fields (different from $\mathbb{F}_2$). We moreover show that Borel subgroups of Kac-Moody groups over local Bézout domains also have a presentation as an amalgamated product of their rank 2 subgroups as soon as this holds for their rank 3 subgroups. In particular we deduce that Borel subgroups of simply laced Kac-Moody groups over a local Bézout domain (with residue field different from $\mathbb{F}_2$) have such a presentation if and only if they are of 3-spherical type.

Rota-Baxter Lie groups are the group-level counterparts of Rota-Baxter Lie algebras of weight $1$. In this paper, we first establish a correspondence between representations of Rota-Baxter Lie groups and representations of Rota-Baxter Lie algebras of weight $1$ via differentiation and integration. We then develop a cohomology theory for Rota-Baxter Lie groups with coefficients in a representation by constructing an explicit cochain complex in arbitrary degrees. We justify this theory by proving a van Est type theorem, which relates the cohomology of a Rota-Baxter Lie group to that of its Rota-Baxter Lie algebra of weight $1$. As an application, we prove that equivalence classes of central extensions of a Rota-Baxter Lie group are classified by the second cohomology group.

Motivated by the EPPA problem for finite tournaments,we consider various extension properties for partial permutations. We show that for any set $\Pi$ of prime numbers, the $\Pi$-extension property is equivalent to the $\Pi$-LERF. As a consequence of known results, the odd-extension property is then equivalent to the EPPA for finite tournaments. To study the $\Pi$-extension property, we reformate the property using the concepts of EP problems and $\Pi$-solutions. We then show that the existence of a $\Pi$-solution for an EP problem depends entirely on its fundamental group. In particular, when the fundamental group is trivial or cyclic, the EP problem has a $\Pi$-solution for any $\Pi$. These extend some known results of Huang, Pawliuk, Sabok and Wise [HPSW19]. We give examples of EP problems without nilpotent-solutions; they witness that the $p$-extension property and the $p$-LERF fail for any prime $p$. Then we consider a special kind of EP problems whose fundamental groups have two generators. We show that any 1-weakly wandering problem has a $\Pi$-solution for any $\Pi$. For 2-weakly wandering problems, we show that they all have odd-solutions and we completely characterize those without nilpotent-solutions.

Abel's classical theorem shows that a continuously differentiable one-dimensional associative law with a two-sided unit is locally equivalent to addition under a change of coordinate. We prove an $n$-valued analogue in the algebraic and formal settings. Over a field of characteristic zero, every associative algebraic $n$-valued monoid on $\mathbb P^1$ with a strong two-sided unit is commutative. Over a reduced commutative $\mathbb Q$-algebra, any one-dimensional formal $n$-valued group is commutative. We then classify $n$-valued formal groups over arbitrary $\mathbb Q$-algebras that are integral domains: for any $n$-valued formal group there is a unique divisor $m$ of $n$ such that the law is strongly isomorphic to the $n/m$-fold diagonal of the cyclic elementary $m$-valued law $\mathbb G_m$. This proves Kholodov's classification conjecture over integral domain $\mathbb Q$-algebras.

We construct an explicit infinite virtually free group $G=C_4\ast_{C_2}D_3$ with a finite inverse-closed generating set whose Cayley graph is $2$-geodetic but not geodetic. More precisely, we describe the full language of geodesics and show that every element of $G$ has a unique geodesic representative except for a single element, which has exactly two. This gives a counterexample to the conjecture that an infinite group admitting a $k$-geodetic Cayley graph must admit a geodetic Cayley graph with respect to the same generating set. We further prove that $G$ is freely indecomposable and hence non-plain. Consequently, Shapiro's conjecture that every geodetic group is plain and the conjecture that every hyperbolic $k$-geodetic group is geodetic cannot both hold.

Let $\pi:G\to \mathcal{U}(L^{2}(G/P,\nu))$ be the boundary representation of a non-compact connected semisimple Lie group $G$ with finite center on its Furstenberg-Poisson boundary $(G/P,\nu)$. Let $\Gamma\subset G$ be a lattice in $G$ (uniform or not). We show that for any continuous functions $\varphi,\psi,\varphi',\psi'$ on $G/P$, $$ \lim_{n\to\infty}\frac{1}{|\Gamma_{n}|}\sum_{\gamma\in \Gamma_{n}}\frac{\langle\pi(\gamma)\varphi,\psi\rangle\overline{\langle\pi(\gamma)\varphi',\psi'\rangle}}{\Xi^{2}(\gamma)}=\langle \varphi,\varphi'\rangle\overline{\langle\psi,\psi'\rangle}, $$ where $\Gamma_{n}$ is a ball in $\Gamma$ with radius $n$ relative to a natural length function and with center the identity element of $G$, and $\Xi$ is the restriction to $\Gamma$ of the Harish-Chandra function of $G$. As a corollary, we deduce that when the real rank of $G$ is one, then the analogous convergence holds for any $\varphi,\psi,\varphi',\psi'$ in $L^{2}(G/P,\nu)$ if and only if the lattice $\Gamma$ is uniform.

We determine the torsion subgroup of $H_{m-1}(\mathbb{B}_mK_{m+1,m+r-1};\mathbb{Z})$ for $m\ge2$ and $r\ge0$: top homology with arbitrary coefficients is the kernel of an unsigned subset-inclusion matrix, and its integral diagonal form determines all primary summands. Every finite order occurs, with explicit representatives. Generalized theta classes span an embedded copy of the cokernel of the inclusion matrix, containing all torsion; for $r\ge m$ they generate the torsion, each of order $\operatorname{lcm}(1,\ldots,m)$. For every prime power $q$ and $m\ge q$, the graph $K_{m+1,m+q-1}$ is minimal in the minor order for order-$q$ torsion in $H_{m-1}(\mathbb{B}_m)$. In particular, odd torsion first appears in $H_2(\mathbb{B}_3K_{4,5})\cong\mathbb{Z}^{155}\oplus(\mathbb{Z}/2)^4\oplus\mathbb{Z}/3$, and no proper minor of $K_{4,5}$ has odd torsion in $H_2(\mathbb{B}_3)$. For arbitrary part sizes, we give a multiplicity-free decomposition of $H_m(\mathbb{B}_mK_{a,b};\mathbb{Q})$ under vertex permutations and prove that $H_{m-1}(\mathbb{B}_mK_{a,b};\mathbb{Z})$ has no $p$-primary torsion when $a,b\ge2m-1$ and $p\ge m$ is an odd prime. The explicit order-$q$ class retains its order under every enlargement of the second part of the graph, while for $m=q=p$ an odd prime it is killed by a specified enlargement of the first part.

For a set $\mathcal P$ of odd primes, let $\operatorname{CT}_{\mathcal P}(\mathbb{Z})$ denote the group generated by all class transpositions whose moduli have no odd prime divisors outside $\mathcal P$. We prove that \[ \bigl\langle \operatorname{CT}_{\mathcal P_1}(\mathbb{Z}),\operatorname{CT}_{\mathcal P_2}(\mathbb{Z}) \bigr\rangle = \operatorname{CT}_{\mathcal P_1\cup\mathcal P_2}(\mathbb{Z}) \] for any sets $\mathcal P_1$ and $\mathcal P_2$ of odd primes. This gives a negative answer to Question 21.75 in the Kourovka Notebook.

How far from finite can an amenable group be? Every finite piece of an amenable group can be imitated by permutations of a finite set, and Cornulier's sofic profile counts how many points such an imitation needs at accuracy $1/r$. We build a finitely presented elementary amenable group with a finite piece whose profile is $\exp(r^{1+o(1)})$, close to the most our counting method can ever detect. The construction is a lamplighter with its lamps on configurations rather than on points. Houghton's group moves the points of three rays, finitely supported affine maps act on their binary configurations, and a copy of $S_3$ sits on every configuration. Then $N$ points carry $2^N$ lamps, yet any two lamps can be brought together by moves that each involve at most three points. The group embeds in Brin's group $3V$, so $3V$ too has a finite piece of nearly exponential profile. The group also gives an explicit quantum channel on $\mathbb C^{873}$ that is cheap to use once and expensive to use many times. A device that applies it $n$ times, releasing each output before the next input arrives, needs about $n/B$ qubits of memory when it may spend $B$ bits of purity, and exact devices achieve this up to polylogarithmic factors. With exchange at the optimal rate, the least memory is about $\sqrt n$, attained with purity of the same order. The channel is factorizable and lies in the closure of channels with finite maximally mixed baths, yet any such bath that imitates it to accuracy $u$ needs dimension $\exp(u^{-1+o(1)})$, although its minimal Stinespring dilation implements it exactly with a pure environment of dimension $130$. The engine is a one-round theorem: after a single use, the memory of any such device carries an approximate representation of the group.

Let $p$ be an odd prime and $H$ a finite $p$-group. Hertweck and Soriano parametrize the central Frattini extensions of $H$ by subspaces of a relation module $V$, matching isomorphism classes of extensions with orbits of $\mathrm{Out}(H)$ and isomorphism classes of the associated small group rings with orbits of $\mathrm{Out}(\mathbb{F}_p H)$. The discrepancy between the two is measured by the group $U(H)\leq GL(V)$ of displacements induced by the normalized unipotent automorphisms of $\mathbb{F}_p H$; it is the mechanism behind the counterexamples to the modular isomorphism problem in characteristic $2$. We prove that for $H$ of exponent $p$ and class at most $p-2$ every such automorphism acts on $V$ exactly as an automorphism of $H$ does, and that $U(H)=\exp\Omega\bigl(\mathrm{Der}_{>0}(\mathfrak{l}(H))\bigr)$, where $\mathfrak{l}(H)$ is the Lazard Lie algebra of $H$, $\mathrm{Der}_{>0}$ its derivations raising the lower central filtration, and $\Omega$ the natural action of derivations on $H_2(\mathfrak{l}(H))$. Hence $U(H)=1$ precisely when every derivation of positive degree acts trivially on $H_2(\mathfrak{l}(H))$. We compute $U(H)$ for all groups of class $2$ and exponent $p$, obtaining $p^{2n(n-1)}$ for the Heisenberg groups $\mathrm{Heis}(p^n)$.

We study the QWEP property of twisted crossed products and its relation to Fourier multiplier transference on noncommutative $L_p$-spaces. We show that QWEP is not preserved by crossed products of QWEP von Neumann algebras by hyperlinear groups. Our examples include trace-preserving actions of a torsion-free residually finite Kazhdan group on a diffuse abelian algebra and on the hyperfinite type II$_1$ factor. We also obtain non-QWEP crossed products with nontrivial scalar cocycles. For scalar twists, we prove that a finite trace-preserving crossed product is QWEP if and only if its ordinary crossed product and its twisted group von Neumann algebra are QWEP. In the positive direction, we establish a restriction theorem for scalar twists of regular wreath products. These results provide QWEP inputs for Fourier multiplier transference and identify a completely bounded multiplier norm obstruction to QWEP. Finally, operator-valued cocycles give finite QWEP crossed products whose acting groups are not hyperlinear.

In this paper we consider groups of complex hyperbolic isometries generated by three complex involutions. Our first main result states that if the group is discrete and at least two of the mirrors of complex reflections intersect, then the group may be put into a normal form involving the ordering of traces of short, even length words. Our second main result refines a conjecture of Schwartz (proved by Grossi) that determines the type of the group in terms of the angles between the mirrors. That is, in the parameter space of complex triangles with given angles, it determines which of two given words becomes elliptic first.

We study the persistence of non-planarity in boundaries of hyperbolic groups under quotients, focusing on quotients by large enough powers of infinite order elements. We prove that the non-planarity of the boundary of a one-ended hyperbolic group is preserved under such quotients. As a consequence, we obtain an analogous result about the persistence of Menger curve boundaries.

For a closed orientable surface of genus at least 2, we study the large-scale geometry of the (symmetrized) Lipschitz metric on its space of projective filling geodesic currents. We show that the rank and, hence, asymptotic dimension of the space of filling geodesic currents is infinite. This gives the first example of a natural proper space associated to a closed surface, with a metrically proper isometric action of the mapping class group, that has infinite asymptotic dimension. In contrast, we show that any finite simplex of geodesic currents has rank uniformly bounded above by $8g-9$. We then show a sufficient criterion for its rank and asymptotic dimension to be equal to $8g-9$ and construct examples satisfying it, thus showing the bound is sharp. We further prove that Teichmüller space equipped with the Lipschitz metric has rank equal to $3g-3$. In particular, the space of projective filling geodesic currents and the Teichmüller space are not quasi-isometric.

Let $G$ be a finite group and $x \in G$. In 2013, Hai-Reuven proved that the cardinality of the centralizer of $x$ divides the cardinality of the solubilizer of $x$. Settling a conjecture of Mousavi, Poozesh and Zamani, we show that the cardinality of the normalizer of $\langle x \rangle$ is also a divisor. We actually give direct short proofs of more general results.

A group is conjugacy separable if every pair of non-conjugate elements remains non-conjugate in some finite quotient. Virtually polycyclic groups, and hence in particular all finitely generated nilpotent groups, are conjugacy separable. The conjugacy separability growth function measures the complexity of distinguishing non-conjugate elements in finite quotients by giving the smallest order of a finite quotient $Q$ that separates them. Recent work showed that this function admits polynomial upper and lower bounds for nilpotent groups, but these estimates are neither explicit nor optimal. We prove improved polynomial upper bounds whose degree is at most linear in the nilpotency class and at most quadratic in the Hirsch length. For $2$-step nilpotent groups, we obtain sharper estimates and show that these estimates are optimal among bounds depending only on the nilpotency class and Hirsch length. This gives a correction of an error in the existing literature. The main tool is a translation of conjugacy separability growth into Lie rings, generalizing previous work in the case of residual finiteness.

A flag complex $L$ determines a locally CAT(0) cube complex $P_L$ with the links of all vertices isomorphic to $L$. The fundamental group of $P_L$ is the commutator subgroup of the right-angled Coxeter group $W_L$. We observe that an edge contraction $L \rightarrow L/e$ determines a $2$-fold branched cover $P_L \rightarrow P_{L/e}$ (up to a homotopy equivalence). We use this branched cover to give an inequality between the $\mathbb{F}_2$-$L^2$-Betti numbers of $W_L$ and $W_{L/e}$. This reduces an $\mathbb{F}_2$-version of the Singer Conjecture for right-angled Coxeter groups to minimal flag triangulations of even dimensional spheres. We also use this inequality to compute the $L^2$-Betti numbers of $W_L$ when $L$ is a flag triangulation of the projective plane or the torus.

Given a group $G$, the prime graph of $G$, denoted $\Gamma(G)$, is the graph with vertex set the set of primes $p$ such that there exists an element of order $p$ in $G$, and such that there is an edge between primes $p$ and $q$ if and only if there exists an element of order $pq$ in $G$. Let $T$ be nonabelian finite simple group. A $T$-solvable group $G$ is a group such that there exists a finite subnormal series of $G$ such that each factor is either abelian or isomorphic to $T$. In this paper, we continue the classification of the prime graphs of $T$-solvable locally torsion groups and classify prime graphs of locally torsion metanilpotent groups using tools from model theory. We also classify the prime graphs of ultragroups with certain first-order properties and prove results on the number and type of certain classes of groups that can realize a given graph as a prime graph.

We introduce SemiBase, a project that computes and formally certifies finite identity bases for small semigroups. Deciding finite basability is undecidable for finite algebras and remains open for finite semigroups. The task requires a proof that a candidate basis is complete, or a proof that none exists, rather than a single first-order validity query. LLM-guided agents search for these proofs; a referee agent rebuilds them from source, and the Lean kernel checks the resulting corpus in a final audit. Humans choose targets and approve final outcomes. We certify every semigroup of order at most 6: all 1309 semigroups of order at most 5 and all 15973 of order 6, including proofs that the four known nonfinitely based semigroups have no finite basis. The bases for order 6 define 505 distinct varieties, whose inclusion order Vampire determines except for four pairs. The resulting catalogue is a machine-checked account of results scattered across the literature and a tested foundation for order 7.

The prime graph of a finite group $G$ is the graph $\Gamma(G)$ with vertex set the set of prime divisors $\pi(G)$ of $|G|$ and an edge between vertices $p, q\in\pi(G)$ if and only if there exists an element $g\in G$ with order $o(g) = pq$. Given a finite nonabelian simple group $T$, a group $G$ is $T$-solvable if there exists a composition series of $G$ such that every composition factor is either abelian or isomorphic to $T$. In this paper, we introduce the prime graph of an infinite group, the graph $\Gamma(G)$ with vertex set $\pi(G) = \{o(g):g\in G\text{ and }o(g)\text{ is prime}\}$ and an edge between $p,q\in\pi(G)$ if and only if there exists an element $g\in G$ with order $pq$, and generalize several results on the prime graphs of finite solvable and $T$-solvable groups to results on the prime graphs of members of certain classes of infinite solvable and $T$-solvable groups.

We show that, unlike the lower central series, the dimension series of a Lie ring need not stabilize when two consecutive terms coincide; in fact, arbitrarily long finite plateaux occur. We prove that $D_n(L)/\gamma_{n+1}(L)$ is central in $L/\gamma_{n+1}(L)$ for every $n\geq1$, in contrast to the group case. If $c\geq2$ and $L$ is metabelian and nilpotent of class at most $c$, we prove that $D_{2c-1}(L)=0$. In particular, our results imply that $D_5(L)\subseteq\gamma_4(L)$ for every Lie ring $L$.

In this paper we prove a generalized version of Mel'nikov's Criterion, which has also been proved by Haran, and use it in order to connect the realization as absolute Galois groups of free pronite and free pro-p products of a sheaf of pro-p groups.